18.306-MIT. Advanced PDE with Applications. Spring 2018. Tue and Thu 9:30--11:00. Room 2-147 Rodolfo R. Rosales. Brief points/topics for the Lecture Summaries. The Lecture Summaries file refers to these #nnn points. [PSQ] means Problem Set Question. [PTS] means Plan to Type Soon [soon being relative]. Things like [S16] and [003] are the keys to a topic in the "Lecture topics for 306" notes. Things like #S04 and #005 are for topics not yet typed. % ========================================================================== % % -------------------------------------------------------------------- #001 ODE's and PDE's ODE solution: determined by a set of constants. Examples (#002 below). PDE solution: determined by functions. Examples (#003 below). ODE: Initial value and boundary value problems. Existence-uniqueness theorem for ODE IV problem. No analogous theorem for PDE's. Closest is the Cauchy-Kowalevski theorem, which requires very strong restrictions (i.e.: analytic functions and restrictions on the derivatives --- order in space cannot exceed order in time). % % -------------------------------------------------------------------- #002 Some facts about ode's: Standard form of n-th order ode [dY/dt = F(Y, t)]. General existence, uniqueness, etc. theorem for ode IVP exists. IVP is well posed ... define ill/well posed. Physical meaning/importance of being well posed. General solution depends on n constants. Examples and proof [use well-posedness of the IVP for ode's]. Boundary value problems for ode's. Example: think of determining a hanging chain shape. Some BVP have a unique solution, others may have no solution, or many, even infinitely many. Example: d^2u/dx^2 + pi^2*u = 0 on 0 < x < 1, u(0) = 0, u(1) = 0, has infinitely many solutions. Example: d^2u/dx^2 + u = 0 on 0 < x < 1, u(0) = 0, u(1) = 1, has only one solution. Example: d^2u/dx^2 + pi^2*u = 0 on 0 < x < 1, u(0) = 0, u(1) = 1, has no solutions. Completeness of IVP ode theory reflected in numerics: IVP solvers solid and robust. BVP solvers much less so. % % -------------------------------------------------------------------- #003 Simple facts about pde's: Define what pde is. There is no standard form, nor any general well-possedness theorem. PDE solutions include free functions, not constants. Simple example: u_t = 0 General sln u = f(x). Another simple example: u_t + u_x = 0 General sln u = f(x-t). [X] [X] Find solution by showing solution is a constant along lines x = t + const. (characteristics). Simplest example of solution by characteristics. Along the characteristics PDE yields an ODE. Another example: u_t + c*u_x = a*u ............................. [PSQ] Use the "trick" above (method of characteristics [#1]) to find the general solution. Then study which BV problems make sense: In these cases, the problem is well posed: a solution exists, it is unique, and depends continuously on the data. First look at causality: BV problems that do not make physical sense because they violate causality. [#1] We will develop this method in detail later in the course. % % -------------------------------------------------------------------- #004 Ill posed PDE problems. EXAMPLE. Thermal archeology. Can you recover the temperature in the past from today's data? Answer: No. Consider the IVP for backward heat equation, u_t + u_xx = 0, with (say) u periodic of period 2*pi and u(x, 0) = U(x) given. Then study what the effect on the solution that high frequency, small perturbations, to U cause. Note: Will derive heat equation (for a thin, insulated, wire) using conservation of heat and Fick's law of difussion later [see #008]. EXAMPLE: An engineer's impossible dream. Would this not be nice? Can you recover the steady state temperature inside a body from knowledge of the temperature and heat flux along some part of the boundary? Answer: No. Consider the steady state heat equation (Laplace equation) on a square, with the temperature and flux given on one side, zero temperature on the two adjoining sides, and nothing known about the opposite side. Then study what high frequency, small perturbations, to the data cause .................................... [PSQ] % % -------------------------------------------------------------------- #005 Well and ill-posed problems. Why is this important. Examples: see #004. (A) In these examples the growth rate of perturbations goes to infinity as the frequency grows. There can be no control over the errors, unless a frequency cut-off for the allowed perturbations occurs. See (C). (B) You may say: well, but these are silly examples. But, note: B1. Easy to be smart after someone finds the answer. Often it is not so clear that the question is ill-posed. See #006. B2. There are many (current) models for various physical phenomena known to be ill-posed under some conditions, but it is not known why/how to fix them. Something is wrong and/or missing from the models. Examples in: multi-phase flows, continuum phase transitions models, square wave model for detonations. (C) Possible Fix: filtering --- get rid of the high frequencies. Works if filtering makes sense within the problem context. It can also lead to nonsense if applied mechanically. Examples: C1. CAT scan. Point reconstruction of the image not possible. Do away with this. Get only local averages (convolution of answer with a filter kernel). C2. Image reconstruction. Similar to C1; cannot ask for too much. C3. Solving the Poisson equation with \ @#@ SKIP for now. interface conditions. CFM / May see later. (D) IMPORTANT. Difference between ill posed with arbitrarily large growth rates, and sensitivity to initial conditions in chaotic dynamics. (E) An example from ode. \dot{y} = F(y) with F not Lipschitz does not mean automatically bad. But it might: Cylindrical water bucket with small hole at the bottom, immersed into a large pool up 1/2 its depth. Let then h be the water level difference, surface inside to surface outside [large pool: neglect changes in surface outside]. Then, it can be shown [s] ................... \dot{h} = -sign(h)\sqrt{|h|} [s] Changes are slow: use Bernoulli to get flow through hole, as driven by hydrostatic pressure difference. Analyze eqn. Show it is fine forwards in time, but not backwards. Explain why backwards, while ill-posed, is physically correct. % % -------------------------------------------------------------------- #006 % @#@ SKIP now. Do later [#S23]. .................................... [PTS] Another example of an ill-posed problem. Numerics for the Navier Stokes equations and PPE approaches. Describe what happens if the incompressibility condition is replaced by a PPE with a naive B.C., such as what one gets by taking the normal component of the momentum equation evaluated at the boundary. Do the linear example only. % % -------------------------------------------------------------------- #007 Causality: a simple example. Consider IBV problems for u_t + u_x = 0, or u_t + u_x = -u, and check which ones make sense (use the general solution). Imposing causality makes the problem well posed: the solution exists, is unique, and depends continuously on data. In continuum mechanics (and other similar settings) causality follows because equations like the above are only approximately valid, with "small neglected effects" that include dissipation. This dissipation effects act like the heat equation, and can be neglected forward in time only. Why (and if) causality applies at the level of "fundamental" physics is not clear, and has been an intense subjecy of discussion. % % -------------------------------------------------------------------- #008a CONSERVATION LAWS (1-D and n-D) AND PDE Continuum limit. Densities, fluxes and sources. Derivation of pde by conservation laws. INTEGRAL and DIFFERENTIAL forms. --- PDE given by a conserved density and corresponding flux and sources. --- Systems of conservation laws. --- Closure and closure strategies. Quasi-equillibrium. Thermodynamics. Equations of state: Fluxes written in terms of conserved densities. Gives rise to ``hyperbolic'' equations, if not ill-posed (more later). % % -------------------------------------------------------------------- #008b EXAMPLE: Kinematic waves. Traffic flow and river flow. Examine the properties of the flow equations of state for these two cases; plot Q = Q(rho). Point to difference: convex and concave. Discuss how this affects causality. Kinematic equation: rho_t + q(rho)_x = 0; rho = conserved density. Can write as rho_t + c(rho)*rho_x = 0, where c = dq/drho. c has dimensions of velocity ... what is it? It is NOT the FLOW velocity, defined by q = flow rate = u*rho. Note c increasing/decreasing for river flow/traffic flow cases. % % -------------------------------------------------------------------- #008c Examples where causality plays a role. -- Use the traffic flow equations [linearized tunnel problem]. Which end of the tunnel can be used to "control" the traffic? -- River flow equations. Same question for flow control. % % -------------------------------------------------------------------- #008d EXAMPLE: --- string equation. @#@ SKIP for now .................... [PSQ] --- string over an elastic bed. @#@ SKIP for now .................... [PSQ] In general, equations for ELASTICITY --- will come back to this later. @#@ SKIP for now. % % -------------------------------------------------------------------- #008e EXAMPLE: Euler equations of gas dynamics (1-D) and closure via equilibrium thermodynamics. rho_t + (rho*u)_x = 0, conservation of mass (rho*u)_t + (rho*u^2 + p)_ x = 0, conservation of momentum (rho*E)_t + (rho*u*E + p*u)_x = 0, conservation of energy E = (1/2)*u^2 + e, e = internal energy, e = e(p, rho) .... Thermodynamics. % % -------------------------------------------------------------------- #008f EXAMPLE: heat equation for a rod. Conservation of heat and Fick's law. (rho*c*T)_t + (-kappa*T_x)_x = 0, conservation of energy c = heat capacity (per unit mass). rho = density kappa = heat conductivity. If rho*c = constant, can write T_t = (nu*T_x)_x, nu = kappa/(rho*c) = heat diffusivity. Extension to more than 1-D [see #008g] Flux is -kappa*grad(T). Boundary conditions: Dirichlet ... temperature given along boundary. Neuman ... heat flow = kappa*n.grad(T) given along boundary. n = unit outside normal to boundary thus formula is for "heat flow INTO REGION" Robin/mixed. Assume cooling by a fluid ===> heat flow through boundary proportional to (boundary temperature minus fluid temperature) kappa*n.grad(T) = alpha*(T_f-T) alpha > 0 [heat flow in when fluid is hotter]. % % -------------------------------------------------------------------- #008g Derive general conservation law [differential form] in 2-D or 3-D rho_t + div(q) = S rho = density of conserved stuff q = flux of conserved stuff, a vector. q = rho*u, u = flow velocity. S = sources of conserved stuff % SIDE REMARK [on sources and sinks and fluxes] ====================== % % Not covered in the lectures, but students should read this. % % % % The distinction between a source and a sink can, at times, be a bit % % confusing to the student. For example, in classical Newtonian mech. % % a force is always a "source" [i.e.: it produces] of momentum. e.g., % % \dot{p} = force, where p = momentum = mass times velocity % % % % However, in the context of conservation laws, forces sometimes are % % fluxes, and sometimes are sources [fluxes and sources conceptually % % are DIFFERENT]. Why, what is the difference? % % First an example: % % In the Euler equations of gas dynamics conservation of momentum is % % [gas in a pipe with cross-section area A] % % % % (rho*u*A)_t + (rho*u^2*A + p*A)_x = rho*f*A % % where % % rho = gas density (mass per unit volume) % % u = gas velocity % % p = pressure (force per unit area) % % f = body force (force per unit mass) % % an example of f is g = gravity % % Here p*A is a momentum flux % % rho*f*A is a momentum source % % But both are "forces". Why is one a flux and the other a source? % % % % The difference is this: % % % % SOURCE/SINKS ....... arise from external supplies of the conserved % % quantities to the system. In the case of momentum, some applied % % force: % % --- you have a beam, which has weight, so gravity acts on it, or % % maybe you hang something from it. % % --- you have a string made of a conductive material vibrating in % % a magnetic field. This creates currents in the string, which % % then yield forces. % % --- the body forces for the example of the Euler equations above. % % --- in traffic flow, cars enter or leave the highway from feeder % % roads [say, 10 cars per second per mile enter the highway]. % % These are SOURCES. It is conserved stuff joining, or leaving, the % % system from outside. It is not, for example, cars moving on the road % % from one place to another [this is FLOW]. % % % % FLUXES, on the other hand, are due to exchanges of the conserved % % quantity between parts of the system. For momentum: one part of % % the system applies a force on another. In gas dynamics (above) % % % % rho*u^2*A = (rho*u*A)*u is the rate at which momentum flows from % % left to right because of the gas velocity u. At any % % point x_p along the pipe, momentum leaves the side % % x < x_p, and enters x > x_p, at a rate rho*u^2*A. % % p*A is the force, by the pressure, from the gas in x < x_p, % % on the gas x > x_p. Again, this means momentum is being % % lost on one side, and being gained on the other, at this % % rate. % % % % Other examples of fluxes: % % In the Traffic flow problem, across some point along the road, cars % % flow from x < x_p to x > x_p at some rate q = rho*u. % % In a string, one side of the string applies a force on the other ... % % this causes a flux of momentum from one side to the other. % % % % Same thing with any other conserved quantity. For example, energy: % % --- Heat moves from hot to cold, at a rate proportional to the % % gradient of the temperature. This is a flow. Energy moves from % % one part of the system to another. % % --- But you can heat a system, say water in a cup, with microwaves. % % This is a source: some external (to the cup-with-water system) % % set-up supplying energy --- some amount per unit volume. % % --- Typically, forces that show up as fluxes in the conservation of % % momentum, will do work that generate fluxes in the conservation % % of energy. Similarly, forces that show up as sources will do % % work that shows up as a source in the conservation of energy. % % % % Finally, note that something that is a source at some level of % % modeling, can became a flux at another. For example, if I have a % % wire, and supply heat to it along its length from some heating % % element, this will show up as a source at the simplest level. But, % % if I model the heat source as well, and my system is now the wire, % % plus the heating element, it could be that now my source becomes a % % flux [heat flux across the boundary of the wire]. % % % % ==================================================================== % % % -------------------------------------------------------------------- #008h Other pde we will study: Laplace, Poisson, Helmholtz, Eikonal, Navier Stokes equations, etc. Will derive as they occur. % % -------------------------------------------------------------------- #009 Conservation laws: HIGHER ORDER EFFECTS. Transport/Dissipative terms, Dispersion, relaxation, etc. Eqn. of state includes derivatives [e.g.: of the densities]. % % -------------------------------------------------------------------- #009a DIFFUSION/VISCOSITY. Examples --- Heat flow. In general: diffusion. Same as heat, but something else diffuses [e.g.: Salinity]. Flow satisfies Fick's law. C_t = (nu*C_x)_x, C = concentration. --- Preventive driving in Traffic Flow [see also #010]. Correct flow from q = Q(rho) to q = Q(rho)-nu*rho_x. Then rho_t + Q(rho)_x = (nu*rho_x)_x --- Navier-Stokes (corrections to Euler equation of state). Add momentum transfer by viscous forces. Add work done by viscous forces. Add internal energy flow by heat conductivity. In 1-D rho_t + (rho*u)_x = 0, (rho*u)_t + (rho*u^2 + p - nu*u_x)_ x = 0, (rho*E)_t + (rho*u*E + p*u - nu*u*u_x - kappa*T_x)_x = 0, --- Burgers' equation. ............................ @#@ SKIP for now. u_t + (0.5*u^2)_x = u_xx Important in limiting the size of the gradients that can occur, and preventing infinities from developing. @#@ SKIP for now. % % -------------------------------------------------------------------- #009b % @#@ SKIP. % Conservation laws: RELAXATION EFFECTS Example: ionization in gases. e = internal energy. E1 = Equillibrium energy in faster degrees of freedom. Characteristic equillibration time: t1. E2 = Equillibrium energy in slower degrees of freedom. Characteristic equillibration time: t2. Assume t1 \ll t2, with E1 and E2 known in terms of rho and T. At equillibrium e = E1 + E2. However, for intermediate time scales t1 \ll t \leq O(t2), only the E1 modes can equillibrate. Model this by writing e = E1 + E with dE/dt = -(1/tau)*(E-E2), where tau = t2, and d/dt = material derivative. Example: Model drivers response time in traffic flow dq/dt = -(1/tau)(q - Q(rho)); tau = typical time to reach equillibrium ~ 5 sec.; % % ------------------------------------------------------------------- #009c % @#@ SKIP. % Dispersion. Loosely: Time evolution of wave phenomena where the wave speed depends on the wavelength [hence a continuum of possible speeds]. A dispersive pde can be hyperbolic, or not. --- Linear Case. Constant coefficients. 1-D. Fourier series/integral solution of a constant coefficient linear evolution equation. Define dispersive, and meaning: (a) omega = Omega(k) real valued for k real. (b) d Omega/dk = c_p(k) is not a constant, i.e. d^2 Omega/dk^2 \neq 0. Implies breakdown of the coherence needed for localization by the time evolution. Localized initial conditions loose phase coherence and de-localize (disperse). --- Examples (linear): (c) Linear KdV or Airy eqn: u_t + u_xxx = 0. (d) String on elastic bed equation: u_tt - c^2 u_xx + m^2 u = 0. (e) Schrodinger equation: i*u_t = - u_xx + V*u. Derive (d)? Derive (e) from (d) when m large? (Parabolic approximation). % ============================================================== % Expect students to be familiar with these linear wave concepts: For the elementary sln: u = Re(A*e^{i(k*x-omega*t)}). --- Wave length and wave number: lambda = 2*pi/k. --- Frequency and wave frequency: f = 2*pi/omega. --- Phase theta = k*x-omega*t + phase(A) --- Amplitude = mod(A). Dispersion relation: solution iff omega = Omega(k). Phase and group speed. % ============================================================== % Example 1: u_tt - u_xx + u = 0 ... i.e.: (d) with c = m = 1. Solutions u = Re{A*exp[i*(k*x-omega*t)]} where omega^2 = 1 + k^2 = Omega(k) is real valued, with nonzero 2nd derivative. *** This example is also hyperbolic *** --- Generalization to higher dimensions. --- Generalize to nonlinear problems: Nonlinear dispersive wave theory; u = F(theta, A). For a nonlinear dispersive wave, the wave number is a function of the wave number AND of the wave amplitude. Example 2: u_tt - u_xx + V'(u) = 0, where V is convex. Generalizes example 1 (where V = u^2/2) to the nonlinear case. Look for solution u = U(k*x-omega*t, E), where U = U(theta, E) is periodic of period 2*pi in theta and E is a parameter (equivalent to the amplitude parameter in the linear case). Eqn. for U is the same as that of a particle in the potential V: (1/2)*m*(U')^2 + V(U) = E, where m = omega^2 - k^2 and U' = dU/dtheta. Solutions are periodic if E > min V(U), and mass m can be adjusted to make them 2*pi periodic. Hence, get omega^2 - k^2 = m(E). % % -------------------------------------------------------------------- #010 Higher order (diffusion) in the LWR-Model of Traffic Flow. Go back to #007 and clarify the effects of neglected effects. Microscopic source of causality: higher order effects (e.g. diffusion). We already saw this for the heat equation. Also, distinction between: Causality: A physical condition. Cannot change the past: boundary and other conditions should only influence future. Study the effect of a perturbation for u_t + u_x = 0. It moves to the right, thus cannot impose BC on right edge of domain. versus Can one figure out what the past was from knowledge of the present, possible for some problems --- keep in mind that, even if possible, if the model is only approximate (as most models are) the "small" corrections may limit how much of this can actually be done. % % -------------------------------------------------------------------- #011 Simple classification of pde. From simple to complicated: --- scalar, systems. --- 2-D, 3-D, ... --- first order, second order, ... --- linear, semi-linear, quasi-linear, ... --- Hyperbolic, elliptic, parabolic, dispersive. Hyperbolic ~ waves with finite number of speeds. Describe a time evolution. Wave phenomena, with discrete set of velocities. Time reversible. Signals have a finite speed of propagation; localized data remains localized. Dispersive ~ waves with frequency dependent speed [see #009c] Elliptic ~ equillibrium problems. Parabolic ~ transport effects. No bound on signal speed (infinite). Not time reversible. This list is not inclusive. We will be more precise later. % =============================================================== % % =============================================================== % % The points below [i.e.: #Snn] follow, more-or-less, the Salsa % % book. Point #012 follows after these #Snn points. % % =============================================================== % % =============================================================== % % % -------------------------------------------------------------------- #S01 READ CHAPTER #1 of the book by Salsa (for now you may skip 1.5). This is, mostly review of prior material. And, what is not review, we have covered in the lectures. % % -------------------------------------------------------------------- #S02 Begin with section 2.1 Book by Salsa Heat equation: T_t - D \Delta T = f f = 0: Linear, Homogeneous, superposition works. Steady state: Poisson equation. Another example: pressure in incompressible flow. Steady state and f = 0: Laplace Types of boundary conditions: Dirichlet, Neumann, Robin. Explain importance of signs for Robin: wrong sign produces runaway temperature. D_n T + alpha (T-Ta) = 0, with alpha > 0, where D_n = n . grad T n = outside unit normal to the boundary. Ta = Temperature "outside" (given). Models: cooling/heating by a fluid past the boundary. Well posed problems: On a bounded domain, generally initial conditions plus BC's of the types above (single or mixed) yield well posed problems. The parabolic boundary in space-time: \partial Q_T [Boundary + I.C.] of the domain of integration. In infinite or semi-infinite domains conditions are needed at infinity (for example: bounded solution) to get WPP. % % -------------------------------------------------------------------- #S03 Example (in 1-D). T_t = T_xx + F(x) for a < x < b T = T_a = constant for x = a. \ Dirichlet T = T_b = constant for x = b. / T = G(x) at t = 0. a) Write T = U + \phi(x), where \phi = equillibrium, steady state sln. Then U satisfies problem with homogeneous B.C. b) Non-dim, and reduce problem to u_t = u_xx 0 < x < 1, with u = 0 at x = 0 and 1 and u = g(x) for t = 0. c) NOTE THE POSSIBLE SINGULARITY AT t = 0, even with a smooth g, if g(0) or g(1) do not vanish. Get back to this below. % % -------------------------------------------------------------------- #S04 Separation of variables and normal modes. This will require some knowledge of Fourier series. REVIEW if needed. Construct solution to #S03b using normal modes. --- Note point below in #S04a --- Process works and produces a solution as long as g is L^2. --- Solution is smooth for t > 0, even if initial data is not. Justify term by term differentiation of infinite series. --- Solution achieves initial data in L^2 sense for g in L^2. If g better than L^2, initial data achieved in better sense. Discuss relationship between convergence of series and smoothness of the data (including at the boundaries), via integration by parts. Relate this to point (c) in #S03. Process can be generalized to other B.C. Examples: --- Sketch Neumann, periodic, and mixed Neumann-Dirichlet. --- Robin requires some knowledge of Sturm Liouville problems. See #S04b % % -------------------------------------------------------------------- #S04a Not in Salsa. Normal modes approach. --- General idea for u_t = L*u. --- Motivate using ode analogy. --- Example where normal modes do not work. u_t + u_x = 0; 0 < x < 1; u = 0 at x = 0; plus IV. Nice, well posed problem, with trivial exact solution. But normal modes do not work. % % -------------------------------------------------------------------- #S04b When do normal modes work? Self-adjoint, Skew-adjoint, normal. Sturm Liouville -D [p(x)*Dy] + q(x)*y = lambda*w(x)*y, p & w > 0, D = d/dx plus self-adjoint BC. Show L = (1/w)(- D*p*D + q) self adjoint with scalar product = \int f(x) g(x) w(x) dx 1) Properties of eigenvalues: lambda_n --> \infty like n^2. 2) """"""""""""" eigenfunctions: alternating zeros. 3) Discuss conditions under which get a positive definite operator L [no boundary contributions in the intermediate integration by parts to show self-adjoint, plus q \geq 0]. 4) Physical interpretation: high frequencies killed. Property of diffusion: it fights against oscillations, the harder the shorter the wave-length. Explain intuitively. Exemplify points (1-4) with the Dirichlet or Neumann cases for the heat equation [sine or cosine series]. High frequency eigenvalues and eigenfrequencies: Sketch WKBJ argument, to give intuition! IMPORTANT: About numerical computations and eigenvalues. Suppose you have some linear continuum problem, and you discretize it. The resulting matrices will always have eigenvalues and eigenvectors. But: do these have anything to do with the original problem? Answer: not necessarily! Easy to construct examples where this is not so. Knowing that your problem is self-adjoint is one way to know that the answer is yes. For that matter, if you need to compute the eigenvalues of some large matrix, how stable is the process [i.e.: can you control the errors?] Again, answer is: not in general. However, if the matrix is, say, normal then the process can be made stable. % % -------------------------------------------------------------------- #S04c Singularities at corners: (x, t) = (0, 0) and (1, 0). What happens if the IC does not satisfy the BC, or if the IC has a discontinuity (more generally, some singularity)? Solution becomes immediately smooth, but how? Explain: boundary layer appears, with thickness that behaves like sqrt{t}. --- Hard to see this from normal modes [normal modes good for large times, when only a few matter, but not good for short times] --- Will see this later using the Green's function approach. --- Sketch example by dimensional analysis: Semi-infinite line heat equation, with T(0, t) = 0 and T(x, 0) = T0 ................ [PSQ] Dimensional argument ==> T = T0*f(x/sqrt(nu*t)), with f(0) = 1 and f(infinity) = 0. Problem: "Boundary cooling of initially uniform temperature." Intuitive idea: back to "diffusion dislikes non-uniformity" due to Ficks' law. Why sqrt{t}? Singularities have no length scale, so thickness of boundary layer controlled by only length scale in equation: sqrt{nu*t}. % % -------------------------------------------------------------------- #S05 Unique solution to #S03b using integral bounds. Need some assumptions about the solutions, for example: C^2 in space and C^1 in time is enough [can do better than this ... but beyond this course]. Works for other BC as long as they allow the integrations by parts. Basically: same conditions that yield Sturm-Liouville problems. Extend to more than 1-D. Same idea. "Deep" reason behind: Laplacian + B.C. self adjoint & negative definite. Formally: show \dot{y} = A\,y has unique solution using: A is self adjoint & negative definite. Note: proof only requires \leq c* for some constant. [i.e.: operator bounded from above]. For q = , equation implies: dq/dt = 2* \leq c*q. Hence, q = 0 initially implies q = 0 for all times. Proof of: dq/dt \leq c*q, and q \geq 0, and q(0) = 0 ==> q = 0. Let Q = e^{-c*t}*q. Then \dot{Q} \leq 0, and 0 \leq Q, and Q = 0. Now obvious Q = 0. Thus q = 0. % % -------------------------------------------------------------------- #S06 Maximum principles (heat equation) u_t = Laplacian u --- Intuitive/physical reasoning: heat flow away from any local maximum and into any local minimum, destroying both if inside. --- Consider solution u in a region R in space (potato) with boundary B (the potato peel). Let T > 0 be some time. Then: max. u over 0 \leq t \leq T and x in R+B is less than or equal to max. u over (R at t = 0) + (B over 0 \leq t \leq T). That is max. u "everywhere" less than or equal than max u over [initial conditions + boundary]. Strong maximum principle: is the maximum is actually achieved inside, then temperature is constant. ==> If either the IV or BC are not constant, the max and min of T is *not* inside domain. --- Similar statements apply to minimums. --- This principle yields another proof of uniqueness. Proof for the case when the solution is continuous up to the boundary, continuously differentiable in time, twice continuously differentiable in space, can be found in the book by Salsa. Idea of proof: let q = u_t - Laplacian u. If u satisfies the equation, then it satisfies both q \leq 0 and q \geq 0. For a moment, imagine that we had w such that w_t - Laplacian w = q < 0. [A] Then, if w actually had a local maximum anywhere inside the domain [x in R and 0 < t \leq T], then at this point it would be w_t \geq 0 and Laplacian u \leq 0, contradicting [A]. Thus such points cannot happen. The idea of the proof is to construct such an w from u, via w = u - epsilon*t, where 0 < epsilon is small, and then take the limit epsilon \to 0. For details see the book. % % -------------------------------------------------------------------- #S07 Review of fundamental solutions/Green functions for ODE, and the "delta function". Example: u'' + u = f(x), 0 < x < 1, u(0) = u(1) = 0. Elementary theory of distributions, weak derivatives, and test functions. Motivate: derivatives only happen inside integrals in many cases. Examples: densities are defined in terms of integrals! Heaviside function and derivative. The Delta function. Calculate delta(alpha*x) in 1-D using the definition. |alpha| delta(alpha*x) = delta(x) Note: answer seems to contradict "intuition" given by the naive definition of delta being a function with integral one that vanishes for x \neq 0. Note: in 1-D generalized derivative guarantees that the "Fundamental Theorem of Calculus" is preserved: ............................ [PTS] Consider a function f=f(x), and two points a < b. Assume that f is continuous at x=a and x=b. Then let phi be a test functions which: 1. Vanishes outside a < x < b. 2. Takes the value 1 for a+eps < x < b-eps, 0 < eps. Then it is natural to define int_a^b f'(x) dx = lim_{eps \to 0} \int f'(x)*phi(x) dx But then int_a^b f'(x) dx = lim_{eps \to 0} -\int f(x)*phi'(x) dx However, phi'(x) is not zero only in a very small neighborhood of x=a and x=b, so that int_a^b f'(x) dx = lim_{eps \to 0} -f(a) \int_a^{a+eps} phi'(x) dx -f(b) \int_{b-eps}^b phi'(x) dx = f(b) - f(a) ... which is the Fundamental Theorem of Calculus. % % -------------------------------------------------------------------- #S08 Fundamental solution for heat equation. Dimension n = 1. Obtain similarity form of the solution by: Dimensional arguments. Stretch invariance of the problem, and (assumed) uniqueness. For u_t = D u_xx form is u = (1/sqrt{D*t}} f(x/sqrt{D*t}), where f must have integral 1. --- Substitute form derived, get ode and solve. --- Fix free constants by constraint on integral. u = = (1/sqrt{4*pi*D*t}} exp(-x^2/4*D*t), --- Show solution yields delta function as t \to 0 (t > 0). --- Note solution becomes non-zero everywhere for any t > 0, hence "infinite propagation speed". --- On the other hand "the bulk of the heat" stays within a region of size O(\sqrt{D\,t}). Confirms early dimensional arguments. --- Use form of the solution just obtained to show what happens with initial discontinuities, discontinuities in derivative (integrate by parts), ... etc. % % -------------------------------------------------------------------- #S09 Fundamental solution, n > 1 --- see [A] below. --- Again, as t \to 0, get delta in n-D. --- Solution initial value problem. Generally smooth for any t > 0. Derivation by similarity arguments, same as for #S08. -------- % [A] Final formula involves "area" of unit sphere S_{n-1}. ----- % [A] Derivation using tensorial nature of n-dim heat equation. ---- % [A] Get formula for "area" of unit sphere S_{n-1}, any n. ----- % [A] For [A] use the problem: --- Problem series: Point Sources and Green functions. --- problem: Nonlinear diffusion from a point seed. --- Subsection: Example: Green function for the heat equation in R^d (The area of a sphere in d-dimensions). % % -------------------------------------------------------------------- #S09b Special point: Note that Green's function for heat equation is symmetric [1] G(x, y) = G(y, x). Same thing observed for the Green's function for problems like u'' + u = f(x), 0 < x < 1, with u(0) = u(1) = 0. [2] Why? Relate to self-adjoint problems. For [2], computing inverse of self-adjoint operator. For [1], solution to \dot{y} = A y [y = exp(A*t) y0] when A = self-adjoint operator with discrete spectrum only. G = Sum phi_n(y)^* phi_n(x) e^{\lambda_n*t} Finally: expression of Green's function for self-adjoint operator using orthonormal set of eigenfunctions. Poorly convergent [not surprising, given the delta]. % % -------------------------------------------------------------------- #S10 Fundamental solutions for special bounded and semi-bounded domains, with various BC [Dirichlet, Neumann, Periodic]. Exploit symmetry: Reflection principle. Method of images. Do 1D examples [2D in #S15]. Stress role of symmetries to get Green functions for problems with boundary conditions. The method of "images" is based on the idea of "mirror" symmetry. But others can be exploited [e.g.: for Robin]. % % -------------------------------------------------------------------- #S10b Green's functions and weak forms. Show how to write equations in "weak" form, using test functions. Example: heat equation. --- important for finite elements. --- gives clear mathematical meaning to the problems that Green functions solve. Do example of heat equation in a half line, with initial conditions and Dirichlet BC. Use test functions that satisfy the homogeneous BC, and show that the formulation yields back the original problem for a smooth enough solution of the weak problem. % % --------------------------------------------------------- Eliminated #S11 Now in #S07 and #033. This used to be: Green's functions and distributions. The delta function as a distribution. Formula for derivatives of distributions. Derivative of the Heaviside function in the distributional sense The fundamental theorem of calculus is preserved. % % --------------------------------------------------------- Eliminated #S12 Now in #S09. This used to be: Formula for area of sphere using heat equation. % % -------------------------------------------------------------------- #S13 Lecture Topics for 18.306 --- Random walks and Brownian motion in 1-D [S13]. --- Simple stochastic ode and Fokker Planck equation. --- Brownian motion and FickÕs law. % % -------------------------------------------------------------------- #S14 Critical Mass for Fission. From Salsa's book and from problem series "Diffusion and probability" ["Critical mass"]. % % -------------------------------------------------------------------- #S15 Green function in multi-D and methods of images. --- Do generic case. Images NEED NOT be deltas. Examples with Robin boundary conditions. See: Green's functions #04 [Assigned in Problem Set]. [IVP, heat equation in the semi-infinite line]. Symmetry here is: --- if u solves heat equation, then v = u-u_x solves it too. --- if v solves heat equation, then u defined by u-u_x = v and u bounded at infinity solves it too. u = e^{x}*\int_x^\infty e^{-s} v ds Why? 0 = v_t-v_xx = (u_t-u_xx) - (u_t-u_xx)_x so that u_t-u_xx = a(t)*e^x. From boundedness, a = 0. See: Supplementary material. Method of images. Robin BC in an interval. Both in the problem series: Point sources and Green's functions. % % -------------------------------------------------------------------- #S16 Lecture topics for 18306. ---------------------------------------------------------------- #S16a Duhamel's principle [S16]. Also in Salsa's book: 2.8.3 pp. 71-73. ---------------------------------------------------------------- #S16b First look at Boundary Integral Methods [BIM] From Lecture Topics for 18.306. \S Boundary integral methods. \SS Toy example: heat equation in an interval. % % -------------------------------------------------------------------- #S17 Lecture topics for 18306. --- Tychonov's example and global maximum principle [S17]. --- Also in Salsa's book: 2.8.4 pp. 74-76 (skip 2.9 pp. 77-89). % % -------------------------------------------------------------------- #S18 Lecture topics for 18306. --- Nonlinear diffusion equation and finite propagation speed [S18]. --- The porous media equation. See also Salsa 2.10.1 pp 90-92. % % -------------------------------------------------------------------- #S19 Reaction-Diffusion equations. Fisher's equation. Salsa 2.10.2 pp 93-96. Briefly mention: Waves and patterns in reaction diffusion equations. Spiral, bulls-eyes, etc. % % -------------------------------------------------------------------- #S20 Begin with Laplace and Poisson equation. 1-Applications where it arises [elasticity, heat, EM, gravitational potential, incompressible fluids]. 2-Type of B.C. and relationship to heat conduction and elasticity: --- Dirichlet. Prescribed temperature. Rigid clamped edge. --- Neumann. Prescribed heat flux. No stress. --- Robin. Cooling by fluid flow. Elastically clamped edges. 3-In fluids: See #S23 BC for the pressure complicated. Briefly mention PPE reformulations of Navier Stokes. 4-Laplace equation in 2-D and relationship with analytic functions. --- Define analytic. Show it leads to Riemann-Cauchy equations. --- Harmonic conjugate. See the problem: "Laplace equation in 2-D and analytic functions" in the problem series: "Separation of variables and normal modes." 5-Define harmonic: u is C^2 and solves laplacian(u) = 0. % % -------------------------------------------------------------------- #S21 Various lecture notes for 18306. Section: Laplace and Poisson equations - harmonic functions. Subsection: Mean value theorem, etc. --- Poisson equation. Uniqueness. --- Intuition for the mean value theorem. --- The mean value theorem in 2-D. --- Maximum and minimum principle. --- Equivalence of harmonic with mean value theorem. --- Harmonic functions are C^\infty. For n-D versions of all these results, see Salsa's book [syllabus]. % % -------------------------------------------------------------------- #S22a Various lecture notes for 18306. Section: Laplace and Poisson equations - harmonic functions. Subsection: PoissonÕs formula and HarnackÕs inequality. % % -------------------------------------------------------------------- #S22b Derive PoissonÕs formula. See: Point Sources and Green's functions. Problem: Laplace equation in a circle #01. % % -------------------------------------------------------------------- #S22c Various lecture notes for 18306. Section: Laplace and Poisson equations - harmonic functions. Subsection: The fundamental solutions. Plus [A] & [B] below. [A] ................................................................. [PTS] Point out the limitations that the slow decay (or none in 2-D) of the fundamental solutions imposes on the method of images for the Laplace operator [as opposed to the exponential decay in the heat equation, which allows good approximations to a Green's function with only a few terms --- for moderate times]. Not even convergence is guaranteed! [1] To obtain convergence in dimensions 2 and 3, the "infinities" must be subtracted. Illustrate this with f(x) = sum_{n \neq 0} 1/|x-n| (not convergent), which can be re-defined as f(x) = sum_{n \neq 0} (1/|x-n|-1/|n|) (convergent) upon subtracting the "infinity". [2] The approach in (1) can also be used to accelerate convergence, e.g.: f(x) = sum_{n \neq 0} (1/|x-n|-1/|n|-|x|/n^2) + |x| sum (1/n^2), where the second term can be done exactly. However: convergence is always slow (i.e.: no better than 1/n^p, for some p). A similar situation arises in other elliptic problems. For example, Stokes equation. This leads to serious difficulties --- e.g.: lack of good approximations for suspensions at concentrations which are not very small. [B] ................................................................. [PTS] How to determine a unique solution in infinite domain. Solve Laplacian p = f, with f decaying fast at infinite. --- OK to ask p to vanish at infinite if dimension 3 or bigger. --- In 2-D need to ask p ~ C*ln(r) + o(1) as r \to \infty, where C follows from: \int_[r ill posed! #S23d Solution #1: Projection methods. See Appendix, #S23f. #S23e Solution #2: PPE re-formulations. Derive equation for div(u). Example: SR scheme. Use: u_{tangential} = 0 and div(u) = 0 for momentum equation. Neumann for pressure as obtained by dotting momentum eqn. with unit normal n to the boundary. Add lambda term to "pull" n.u towards zero. Show it gives system equivalent to Navier-Stokes. #S23f. Appendix. Fractional methods. Explain the idea behind fractional methods. In particular, for matrices: exp((A+B)*t) = exp(A*t)*exp(B*t) + O(t^2) Strang splitting exp((A+B)*t) = exp(A*t/2)*exp(B*t)*exp(A*t/2) + O(t^3) % % -------------------------------------------------------------------- #S24 Boundary Integral Methods [BIM] --- Example: 2-D Laplace/Dirichlet BC problem. From Point Sources and Green's functions Subsection: "Background: double layer potentials in 2-D." --- Mention single layer potential for Neumann BC problem. % =============================================================== % % =============================================================== % % NEW TOPIC: Hyperbolic equations, and characteristics. % % We do not follow Salsa any longer. % % =============================================================== % % =============================================================== % % % -------------------------------------------------------------------- #012 EXAMPLE. Linear, constant coefficients. u_t + c*u_x = 0 and u_t + c*u_x = a*u. u = f(x-c*t) and u = e^(a*t)*f(x-c*t) % % -------------------------------------------------------------------- #013 EXAMPLE: First order scalar linear pde with variable coefficients. Time evolution in 1-D. u_t + c(x)*u_x = f(x), with u(x, 0) = U(x). Solution by characteristics yields: u = U(s, t) x = X(s, t) where U(s, 0) = U(s), X(s, 0) = s, U_t = f(X). X_t = c(X), Derive equation for X_s and show X_s > 0. Take s-derivative of X_t = c(X), to obtain (X_s)_t = c'(X)*X_s, with IC X_s = 1. Thus, it never vanishes. Hence: x = X(s, t) can be solved for s = S(x, t). Note: for now assume that the coefficients in the equations are nice (say, at least one continuous derivative). Later on we will worry about coefficients that have singularities. In particular: discontinuities raise the issue of what does the equation mean. The resolution of meaning issues often require going back to the physics of the system modeled by the equation. % EXAMPLE: Solve u_t + x*u_x = 0, -1 < x < 1, with u(x, 0) = f(x). Point out: no boundary conditions needed! Repeat for u_t + x*u_x = u. % % -------------------------------------------------------------------- #014 General set-up a*u_x + b*u_y = c*u + d, with a and b functions of (x, y). --- Show it can be reduced to ode along characteristics (this property defines it as a hyperbolic equation). --- Characteristic form of the equation. --- Allowed type of data: solution given along a curve that intersects (transversally) every characteristic in the region of interests once and only once. See [#A] & [#B]. Reduction to ode along characteristics proves (by explicit construction) existence, uniqueness, and continuous dependence on data [at least in a neighborhood of the curve with the data]. [#A] "Only once" needed because the solution is fully determined, along each characteristic, by a single value. [#B] "Transversally" needed because the derivative along each characteristic is determined by the equation. Hence, for arbitrary data, the data curve cannot be tangent to any characteristic. General formalism: Write solution in characteristic coordinates x = x(s, z) \ s = parameter along data curve y = y(s, z) | z = parameter along charact. u = u(s, z) / Then invert: s = s(x, y), z = z(x, y) to obtain the solution. --- IVP for case where y = time. General solution of the initial value problem. --- Examples: a) Linearized traffic flow. Density waves move backwards through traffic. b) Linearized river waves. Flood waves move forward of fluid particles. % % -------------------------------------------------------------------- #015 EXAMPLE: x*u_x + y*u_y = 0, plus data. --- Write equation for characteristics, and solve them. --- Show general solution for y > 0 is u = F(x/y). --- Use it to write solution to problem with data: u(x, 1) = F(x). Show this defines the solution on y > 0. Example: u = e^x on x = 1. --- Use polar coordinates to show solution has the form u = U(theta). --- Let now u = e^x on y = x^2, for x \neq 0. Why do we exclude the origin? Defines solution for y > 0. % % -------------------------------------------------------------------- #016 % SKIP. Students should do this as an exercise ...................... [PSQ] % EXAMPLE: solve x*u_x + y*u_y = 1+y^2, for y >= 1, with u(x, 1) = F(x) % % -------------------------------------------------------------------- #017 DEFINITIONS. Domain of definition: where is the solution defined by the data. Domain of influence: region affected by a point in the data. Domain of dependence: region that affects the solution at a given point. Explain consequences for numerical methods: CFL condition. Numerical domain of dependence must include actual domain. Numerical domain of influence must be included into actual domain. This for hyperbolic problems. Look at case of diffusion, where (for explicit methods) this applies in the \Delta x \to 0 limit. % % -------------------------------------------------------------------- #018 EXAMPLE showing that the solution is not uniquely defined outside the domain of influence of the data = domain of definition. For the example in #015, let u(x, 1) = exp(-2*x^2). % Then, in the "punctured" plane P0 [the plane without the origin (x, y) = (0, 0)] define u1 and u2 by: u1 = exp(-2*x^2/y^2) ................. for all x, y in P0. u2 = exp(-2*x^2/y^2) ................. for y >= 0 and x^2+y^2 > 0. = exp(-3*x^2/y^2) ................. for y <= 0 and x^2+y^2 > 0. Both u1 and u2 are smooth, and solve the equation for the given data, but they are not equal outside y >= 0 and x^2+y^2 > 0. In fact: can construct infinitely many such u's. % % -------------------------------------------------------------------- #019 For linear equations the domain of definition does not depend on the data, just the place where the data is given. For nonlinear problems this is not true. Two simple semi-linear examples follow: Example a: x*u_x + y*u_y = u^2, with u(x, 1) = F(x) Domain of definition depends on F [solution blows up along characteristics when F not zero]. Example b: u_t + c*u_x = u^2, with u(x, 0) = F(x). Solution not defined for all t > 0 along characteristics where F > 0. Point: when singularities arise, math. goes "bad" and modeling hypothesis have to be revised. Made up hypothetical example: If the equation in example b above were to be a model for the growth of the population of a traveling something, where at low density the probability of reproduction is proportional to the meeting probability of pairs, at high density this hypothesis would certainly have to be modified. Better example: Nonlinear Schrodinger equation for laser beam propagation. Equation is semi-linear (though not hyperbolic), and the phenomena of self-focusing yields singularities. Mention other examples where mathematical modeling idealizations lead to singularities in the solutions. % % -------------------------------------------------------------------- #020 Recall: Domain of definition: where is the solution defined by the data. Domain of influence: region affected by a point in the data. Domain of dependence: region that affects the solution at a given point. Implications for where conditions must be given: u_t + c(x)*u_x = 0 in an interval a < x < b. Causality: If c(a) > 0, BC's needed at x = a, and only then. If c(b) < 0, BC's needed at x = b, and only then. Draw characteristics for various example c = c(x). Numerical implications. CFL condition. Generalize method of characteristics to other first order scalar eqn.: --- Semilinear. [DEFINE] --- Quasilinear. [DEFINE] % % -------------------------------------------------------------------- #021 Quasilinear Scalar Equation. Conservation Laws. Example: u_t + c(u)*u_x = 0 and u(x, 0) = F(x). Solution by characteristics: x = c(F(s))*t + s, u = F(s). Characteristics may cross, leading to multiple values. Implicit form of the solutions: u = F(x - c(u)*t). Crossing of characteristics: They cross if (d/ds) c(F(s)) < 0 somewhere, because x_s = 1 + t*F'(s)*c'(F(s)) vanishes somewhere. [#1] Cannot solve for s = S(x, t). Draw x(s, t), as a function of s, for t < t_c and t > t_c First time of crossing, t_c: Smallest t for which 1 + t*(d/ds) c(F(s)) = 0 for some s. Graphical interpretation of the solution. Wave steepening and breaking. Lego-view of the time evolution: Solution by characteristics satisfies conservation. Space-time plot of the multiple valued region (simple case) ... will relate this later to the envelope of the characteristics. Show derivatives blow up along the boundary of the region. Break-down of mathematical model (quasi-equilibrium hypothesis). What happens after wave breaking? Must go back to physics. Note: in linear and semilinear case, characteristics known a-priori, and provide "good" curvilinear coordinate system. Quasi-linear: they depend on solution. [#1] Plot C'(s) versus -1/t, where C(s) = c(F(s)). [#2] Curves where u_x = infty in space-time. Parametric description. Interpretation as envelope of characteristics. Definition of envelope of family of curves, and derivation of equation for the envelope. % % -------------------------------------------------------------------- #022 Continue with: For u_t + c(u)*u_x = 0 and u(x, 0) = F(x). a) Graphical interpretation of the solution by characteristics. b) Crossing of characteristics. Wave steepening and breaking (infinite derivatives). c) Describe region (in space-time) where solution by characteristics is not valid: locus where the characteristics cross and the solution is multiple-valued. For hump-initial conditions: wedge shaped region with a cusp. --- Compute location of tip of wedge: first time and location where solution goes to infinity. Follows from solving: x = C(s)*t + s and 0 = C'(s)*t + 1, where C(s) = c(F(s)). Envelope of the characteristics! To continue the solution beyond the breakdown, and fill in the region in space-time where the characteristics cross, we need to modify the mathematical model, so that it describes the behavior even after wave breaking. Modeling hypothesis that fails is quasi-equilibrium: at breakdown space and time derivatives go to infinity. The quasi-equilibrium assumption breaks, sooner or later, as a breakdown is approached. To figure out what happens we will examine a few simple situations in Traffic Flow and River Flows, observe what actually happens in them after wave breaking, and then we will modify the mathematical model accordingly [this will involve some idealizations, consistent with the continuum approximation]. % % -------------------------------------------------------------------- #023 Simple example in Traffic Flow: Red light turns green (discontinuous initial data). -- Get expansion fan solution by "smearing" discontinuity. -- Argue: not only is this the "correct" mathematical solution, but it is also the one observed in physical situation. -- Meaning of solution and (qualitative) comparison with observations. -- In this case, the equation ``fixes'' the discontinuity in the IV. The ``bad problem'' in traffic flow, leading to steepening and wave breaking, occurs when the density increases forward (so that the characteristic speed decreases). In this example rho decreases across discontinuity. % % -------------------------------------------------------------------- #024 Simple example in Traffic Flow: Green light turns red (discontinuous initial data) on uniform light traffic with density rho_0 Split into 2 mixed (initial values)--(boundary values) sub-problems. --- Ahead of the light. Draw characteristics, show they cross. Observation: what happens is last car through the light, moving at speed u(rho)_0. Observed behavior has a discontinuity. % --------------------------------------------------------- #024a Note: The space-time path of the last car through the light is right in the zone where characteristics cross, and the characteristics on each side converge into it and ``die'' there. This is how crossing of characteristics is avoided. % --------------------------------------------------------- --- Behind the light. Draw characteristics, show they cross. Observation: cars wait till the ``last moment'' to break behind the cars already waiting behind the light. Observed behavior has a rather ``thin'' zone of breaking moving backward from the light, separating the rho = rho_j zone behind the light, from the rho = rho_0 traffic behind. We idealize this thin zone as a discontinuity. % --------------------------------------------------------- #024b Again: observed behavior has a discontinuity, but now the speed at which the discontinuity moves is not as evidently obvious as before. What is the speed? (see #025). % --------------------------------------------------------- % % -------------------------------------------------------------------- #025 Given a discontinuity (shock), what speed should it have? Back to CONSERVATION. Use integral form of conservation law d/dt int_a^b rho dx = q_a - q_b to derive Rankine Hugoniot condition: shock speed = s = [q]/[rho] #025a 1) Reduces to characteristic speed for infinitesimal discontinuities. 2) Graphical interpretation: shock speed = slope of secant line connecting the shock states in the rho-q diagram. 3) Derive also by looking at conservation in shock frame Flux in = (u_-s)*rho_ \ equality leads to Flux out = (u+-s)*rho+ / #025a Apply #025a to the discontinuity in #024b, then show that the analog of #024a applies: % --------------------------------------------------------- #025b The discontinuity path is in the region with crossing of characteristics. The characteristics on each side converge on the path and end there. Thus crossing is avoided. % --------------------------------------------------------- % % -------------------------------------------------------------------- #026 In #024 and #025 we saw examples where discontinuities (*) (*) Such discontinuities are math. idealizations. In actuality there is a ``thin'' zone of rapid change. appear in the traffic density to resolve crossings of characteristics. Discontinuous transitions occur regularly both in Traffic Flow and in River Flow. We propose the following ``augmented'' theory for conservation laws of the form: rho_t + q_x = 0, with q = q(rho). NOTE 1: some restrictions on q = q(rho) are needed for a good theory. We will get to them later. NOTE 2: the fact that, for sufficiently slow change rates, in some physical situation, one can derive a conservation law of the form above *DOES NOT MEAN* that this theory applies! This theory involves additional assumptions about the physical system, which we will address in some detail later. NOTE 3: later we generalize this theory for systems of conservation laws (e.g.: Euler eqns. of Gas Dynamics). % ------------------------------------------------------------- #026a The pde (and the solution by characteristics) applies in regions where the solution is smooth enough (say: continuous partial derivatives) % ------------------------------------------------------------- #026b Simple discontinuities are allowed along curves x = S(t). This means: the solution has continuous partial derivatives on each side, all the way up to the curve x = S(t). NOTE 4: later on we show how to generalize this to solutions that are much less smooth than this. % ------------------------------------------------------------- #026c Across each discontinuity the Rankine Hugoniot jump conditions apply ...................................... dS/dt = [q]/[rho]. This follows from conservation, and guarantees it. This reduces to the characteristic speed for infinitesimal discontinuities. NOTE 5: IMPORTANT. More than one conservation form can be associated with the same pde (and characteristics). Must know the correct physical one to get the true shock speed. Example: u_t + (1/2*u^2)_x = 0 and (1/2*u^)_t + (1/3*u^3)_x = 0 ===> same p.d.e. % ------------------------------------------------------------- #026d PROCESS: Shocks are introduced to avoid crossing of characteristics. For example: at the tip of the region of multiple values (see #021--#022) a shock is born, whose path separates the characteristics arriving from each side (thus preventing the crossing). The shock path is then determined by the solution of the ode provided by the Rankine-Hugoniot condition ............................... dS/dt = [q]/[rho], where the needed values for rho on each side are provided by the characteristics that arrive into the shock locus from each side (and connect it to the initial conditions). Since the characteristics then terminate at the shock, the crossing and multiple values are avoided. % ------------------------------------------------------------- #026e IMPORTANT: to obtain a well posed model, a RESTRICTION ON THE ALLOWED DISCONTINUITIES (shocks), IN ADDITION to the RANKINE HUGONIOT CONDITIONS, IS NEEDED: The LAX ENTROPY CONDITION. Conservation not enough: allows unphysical solutions. See #028 % ------------------------------------------------------------- #026f The ENTROPY CONDITION is related to this fact: Shocks introduce causality (time arrow) and irreversibility. Solutions without shocks can be run backwards in time (till the characteristics cross in the past). Solutions with shocks cannot. We get back to these points in detail later [#029-#030-#031]. But first some examples. % % -------------------------------------------------------------------- #027 RAREFACTION FANS. Lesson from example: Traffic flow, red light turns green. Provided a first look at how to deal with discontinuities in the data that yield "gaps" in the characteristic field: Smear data at discontinuity. Examples: % ------------------------------------------------------------- #027a u_t + u*u_x = 0, -inf < x < inf and t > 0, with u(x, 0) = 2 for x > 0 and u(x, 0) = 1 for x < 0. % ------------------------------------------------------------- #027b u_t + u*u_x = 0, 0 < x < inf and t > 0, with u(x, 0) = 3 for x > 0, u(0, t) = 2 for 0 < t < 1, u(0, t) = 1 for 1 < t < inf. % ------------------------------------------------------------- #027c u_t + u*u_x = -u, -inf < x < inf and t > 0, with u(x, 0) = 1 for x > 0 and u(x, 0) = 0 for x < 0. % % -------------------------------------------------------------------- #028 Simple examples in Traffic Flow: Red light turns green (discontinuous initial data). -- Solution with a discontinuity allowed, but NOT stable. -- Not that discontinuity generates information. Causality is lost: solution should be generated by the data only. -- First look at Lax entropy condition: discontinuous solution fails to satisfy it. % % -------------------------------------------------------------------- #029 ENTROPY CONDITIONS AT SHOCKS. First peek at this was in #024a and #025b. % ------------------------------------------------------------- #029a Formation argument. Introduce shocks to knock out multiple-valued regions ONLY. --- When do we need shocks? --- When does wave breaking occur? --- Characteristics must cross/converge. HENCE: c- > s > c+ <--- "Lax entropy condition". % ------------------------------------------------------------- #029b Well posed argument. Inspect the various scenarios for characteristic patterns near a shock. See which one leads to well posed problems on each side of the shock, plus determination of the shock path. Only c_- > s > c_+ works (Lax entropy condition): yields 2 problems (one per shock side), each fully determined by the initial data, plus one equation to determine the shock path. % ------------------------------------------------------------- #029c High order effects argument [see also #039 & #040]. Zero viscosity limit; shocks as internal layers. - Assume more detailed model, incorporating the physics that is important in the shock layer. The shocks should occur in this model solutions, in the limit where high order effects vanish. - Example: zero viscosity limit. When the higher order effect is diffusion/viscosity. These effects cause a flow against the gradient (Fick's law), which fight steepening & become larger the steeper the gradient. These effects eventually balance wave breaking, in a ``thin'' layer with a sharp transition. WILL GET BACK TO THIS LATER, IN MORE DETAIL. - WARNING: not all cases lead to shocks; some high order effects do NOT produce shocks. Examples: 1) Zero dispersion limit; e.g.: u_t + (0.5*u^2)_x = d*u_xxx with d \to 0. 2) ``Collisionless'' shocks in plasmas. 3) Waves in a flexible pipe filled with an incompressible fluid (e.g.: garden hose) when the elastic effects along the pipe dominate over bending dissipation. 4) Modulation equations for high frequency waves. For a shock a sharp, thin, transition region is needed. If, for example, the formation of steep gradients triggers waves that radiate away from the breaking region [1-3] above: cannot use shocks. In some cases the multiple solutions even make sense and must be kept [eg.: 4 above] even if the equations may have to be modified after they arise [e.g.: modulation for nonlinear waves]. % ------------------------------------------------------------- #029d Consistency: The above (Lax entropy) works out provided that the Rankine Hugoniot jump condition is consistent with c- > s > c+. Examples: Traffic Flow, concave Q: consistent iff rho- < rho+ (*) River Flow, convex Q: consistent iff rho- > rho+ (*) Show graphs, with shock speed as slope of secant line. (*) Note that this agrees with actual observations. When Q = Q(rho) is neither convex nor concave, things get complicated. Show some plots. - The problem is that Q need not stay on only one side of the secant line. For kinematic waves reasonable resolutions of the difficulty are possible in simple examples. See #032 But the analogous problem (lack of convexity) leads to open, unsolved, problems for systems of conservation laws. We will come back to this later, when we look at systems. % % -------------------------------------------------------------------- #030 ENTROPY CONDITIONS and TIME ARROW. IRREVERSIBILITY. Reason for the name ["Lax entropy condition"]. - Entropy: recall "quasi-equillibrium" is thermodynamics in gas dynamics. The entropy condition in this case (will see this later) is equivalent to 2nd law of thermodynamics: Entropy must not decrease as fluid particles cross shock layer. - Lax: generalization of condition to systems other than Gas Dynamics. Augmented system is irreversible: Once shocks form, cannot go back in time. Characteristics end at shocks, and the info they carry is LOST. Second law of thermo. % % -------------------------------------------------------------------- #031 % SKIP in lectures. Main point here assigned in a ................... [PSQ] % Shocks and dissipation; measuring "information contents". Entropy inequalities for the quasi-linear scalar case. In the case of Gas Dynamics there is a well identified quantity that has to drecrease (entropy, get to this later). Let us see if the same happens for the quasi-linear scalar equation rho_t + Q(rho)_x = 0, with shocks/entropy conditions as above. % ------------------------------------------------------------- #031a Argue that the "information" content in the graph of a function rho = rho(x, t) is given by its "wiggliness". For a given fixed area, the more variation in the function, the more information. Example. Constant: just one number characterizes it. Example. Sinusoidal. Need mean, amplitude, frequency and phase. % ------------------------------------------------------------- #031b How to "measure" the information? Propose 0.5*rho^2 as "information density". As the wiggles increase, for a given area, the integral of 0.5*rho^2 goes up. More generally, any strictly convex function f(rho) can serve as a measure of information density. % ------------------------------------------------------------- #031c Consider now rho_t + Q(rho)_x = 0, with a convex/concave flow rate Q. Then: - For smooth solutions, information conserved, e.g. (rho^2/2)_t + h(rho)_x = 0, where h = INT c(rho)*rho - Consider d/dt int_a^b (rho^2/2) dx, with a shock somewhere between a and b, and compute the contribution from the shock. Show: *information lost* iff *Lax Entropy conditions apply*. Do the example Q(rho) = (1/2)*rho^2. General case in .............. [PSQ] - Connection with high order effects and viscosity: Will show later: Lax Entropy condition is equivalent to the shocks being the limit nu --> 0 for rho_t + q_x = nu*u_xx, nu > 0. For this equation, easy to see that d/dt (int rho^2) < 0. % % -------------------------------------------------------------------- #032a Kinematic equations with a non-convex flux .......................... [PTS] Example: u_t + q_x = 0, with q = -(1/2)*u^2 + (1/4)*u^4 = -(1/2)*u^2*[1-(1/2)*u^2]; c = - u + u^3 = - u*(1-u^2); q is not convex. Let the initial data be: u(x, 0) = alpha*sign(x). -- alpha > sqrt(2) yields steady state shock. -- 0 < alpha < \sqrt(2) yields 2 shock waves connected by rarefaction. Simple physical example: flood waves in river with secondary ["25 year", or "100 year", or whatever] bed ......................... [PSQ] % % -------------------------------------------------------------------- #032b Shocks for equations with source terms. Example: u_t + (0.5*u^2)_x = 1. - Study characteristics, crossings and shock formation. - Derivation of the RH conditions, assuming u is conserved. - Entropy conditions. % % -------------------------------------------------------------------- #033a Shocks and weak solutions. - Explain notion of generalized derivative. Examples: Heaviside and delta function, principal value, etc. - The product rule does not work for generalized derivatives, so using the correct conservation form is crucial. - Weak formulation of the equations, including IC and BC. Equivalence with Rankine Hugoniot conditions. - Point out the appropriate conservation form matters for the weak formulation. - Entropy conditions and incorporation into weak formulation using entropy functionals. Shock jump conditions and generalized derivatives. Get shock jump conditions by matching "singular" part in conservation form of the equation. % % -------------------------------------------------------------------- #033b Characteristics as locus of possible weak singularities in solution. Elliptic as equations that do not accept any singularities. Connection with the "principal symbol" of the equation: write equation by replacing derivatives \partial x_j by k_j. This yields a polynomial in the x_j. Example: Definition of hyperbolic and elliptic for scalar second order pde in 2-D using the symbol. a*u_xx + b*u_xy + c*u_yy + lower order terms = 0 [A] Symbol: S = a*k1^2 + b*k1*k2 + c*k2^2. Elliptic: S \neq 0 for all k1 and k2 not zero. Laplace: S = k1^2 + k2^2. Hyperbolic: S can be factored as two linear polynomials. Wave equation: S = k1^2 - k2^2. See note 3 below. Sketch of argument for weak singularities: Let the curve, G, along which a weak singularity in the solution to [A] can occur be given by the equation phi(x, y) = 0, where phi is some C^2 function defined near G. Let psi(x, y) be a C^2 function defined near G as well, such grad phi and grad psi are not co-lineal. [see note 1] For [A] a "weak singularity" means that u is C^1 near G, C2 away from G, with the 2nd derivatives, possibly, having simple discontinuities across G. [see note 2] Let us now re-write the equation in terms of the local coordinate system provided by phi and psi. Then it can be seen that the equation takes the form (a*phi_x^2 + b*phi_x*phi_y + c*phi_y^2)*u_{phi,phi} = C where C involves u, u_phi, u_psi, u_{psi, psi}, and u_{phi, psi}, but not u_{phi, phi}. The weak singularity assumptions then tell us that C is continuous, while u_{phi,phi} has a simple discontinuity across G. This can only happen if a*phi_x^2 + b*phi_x*phi_y + c*phi_y^2 = 0. Note 1: if the curve is smooth enough, one can always define such a phi (at least near the curve) by: phi is the (signed) distance to the curve from the point (x, y). Then let s be the arc-length along the curve, and define psi = s along the straight line segment normal to the curve, and crossing it, at the point s. Note 2: the idea of a "weak singularity" is that it is a failure in the smoothness of the solution, which is weak enough that it does not destroy the notion of being a solution in the classical sense. It turns out that the exact details of how this is implemented do not matter much. For example: if we assume that the lack of smoothness happens in the third order derivatives, the constraint on the curves across which this can happen is the same. Note 3: Point out relationship of classification names [elliptic, hyperbolic] with the level curves of the symbol. In the particular case of parabolic, note that the symbol [upon a rotation change of variables] corresponds to an ode: a*k_1^2; a \neq 0. Thus the lower order terms matter and one should look at a*k_1^2 + b*k_1 + c*k_2. This is not an ode only if c \neq 0, in which case the level curves are parabolas. % % -------------------------------------------------------------------- #034 Riemann problems and numerical solutions. 1- Introduce Riemann Problem. ** See examples: #035, #036. ** 2- Whole problem is encoded into the Riemann Problem, including the R.H. jump conditions and the Entropy cond. If you can do the R.P., then you have, in principle, everything. If you can show that the Riemann problem is well behaved, then you can be (almost) certain that problem is OK. The solution of the Riemann problem is also the basis for: 3- Godunov-type methods. 3.1 Higher order and generalized Riemann problems. van Leer. 3.2 Higher order and semi-discrete, using only Riemann problem. % % -------------------------------------------------------------------- #035 Riemann problem. Kinematic wave eqn with convex/concave flux ........ [PSQ] Riemann problem for: u_t + Q(rho)_x = 0. [PTS] -- Case Traffic Flow Q concave. -- Case River Flows Q convex. % % -------------------------------------------------------------------- #036 Riemann problem. Conservation law with a point source term .......... [PSQ] Riemann problem for u_t + (0.5*u^2) = delta(x). Give meaning to equation as a conservation law. Physical model: point heat source (laser beam) moving through a gas filled pipe at the sound speed. Point source term at x=0 implies there is a discontinuity there. Appropriate jump conditions needed at x=0, restricted by causality. That is: get Rankine Hugoniot and entropy conditions at x=0. Two derivations: -1 Use "delta functions" to give meaning to the derivatives across discontinuities. This yields the appropriate Rankine Hugoniot jump conditions. Then analyze the characteristics, and use causality and solvability to obtain the appropriate entropy conditions. -2 Smear the delta function and look at the limit when the smearing vanishes. This approach is specially useful to justify the cases where characteristics are parallel to the line x = 0 on the left, while emerging from the right (characteristics can be ``trapped'' inside the delta region). Do some examples of Riemann Problem for the equation. % % -------------------------------------------------------------------- #037 Riemann problem. Kinematic wave eqn. with a flux .................... [PSQ] that is not concave/convex [see #32a and #040a] ..................... [PTS] % % -------------------------------------------------------------------- #038 General description of how to do Riemann problems (preview of what happens for systems). 1- Construct curves in state space (shock and rarefaction branches) that describe the allowed to-and-from (shocks and rarefactions) from a point in state space. 2- Connect states on the right and left of RP going through intermediate states 3- Example: u_t + q_x = 0 for q = q(u) cubic. See #037. Preview for systems: what happens for systems with two variables. % % -------------------------------------------------------------------- #039 Shock structure produced by more detailed physics: Viscosity. See also #029c. Assume that the higher order effect is diffusion/viscosity. This causes a flow against the gradient (Fick's law), which fights steepening and becomes larger the steeper the gradient. Eventually the nonlinear steepening is balanced and a "thin" layer with a sharp transition arises: the shock. Generally expect that 1- Diffusion fights steepening: induces extra flow down gradient. 2- Effects of the large gradients remains local (only where the wave steepening happens). 3- Steeping eventually stopped and a local, thin, transition develops ==> discontinuity in limit. % % -------------------------------------------------------------------- #040 Example: Viscosity solution in Traffic Flow and Entropy condition. Modify flux to q = Q(rho) - nu*rho_x. Justify, explain why. For the modified equation, traveling wave solutions exist and satisfy the shock conditions (both Rankine Hugoniot and Entropy). Why do traveling waves describe what happens near a shock when nu is "small" (intuitively) -- Scales inside the shock layer are much shorter/faster than outside. From shock layer view point, both the shock speed, as well as the "outside" boundary conditions on the left and the right are steady. Hence shock layer should look like a steady traveling profile. DETAILS: look for traveling wave solutions of equation with diffusion. Start with rho_t + c(rho)*rho_x = nu*rho_xx Nondimensional: 0 < nu \ll 1 Shock transition must have width nu (scale t/nu and x/nu) for the higher order effect to balance nonlinear steepening. In this scale (locally) the shock path is a straight line, and the shock profile corresponds to a travelling wave. Hence: Look for solutions of the form u = F((x-s*t)/nu). When do such solutions exist and provide a smooth connection across a discontinuity in the nu --> 0 limit? Problem for F is an ODE boundary value problem. Use convexity and/or concavity of Q = Q(rho) to get a complete theory: Connection exists if and only if the R-H jump conditions are satisfied, and the characteristics converge on the shock path. Use the graphical interpretation of the shock and entropy conditions in the rho-q plane, connect them with the form of the ode for F, and use convexity/concavity: Yields F monotone between the two states % ------------------------------------------------------------- #040a NON-CONVEX FLOWS. See #037 .................................................. [PSQ] and [PTS] Kinematic wave equation rho_t + Q(rho)_x = nu*rho_xx. Examine (graphically) the PROBLEMS that ARISE when Q is neither convex, nor concave. Show that a shock solution may not exist. Shocks are only possible between consecutive zeros of the right hand side of the ode F' = q - s*rho + kappa, which also satisfy entropy. % % -------------------------------------------------------------------- #041 Shocks, dissipation, and "information" contents ..................... [PSQ] This is related to #031. See problem: Zero viscosity limit in scalar convex conservation laws and dissipation. Consider Burgers' equation u_t + u*u_x = nu*u_xx and calculate equation for the time evolution of the information contents E = int (1/2)*u^2. Show it decreases in time due to the nu*u_xx term. Plug in shock layer solution and show that the amount of information loss DOES NOT go to zero as nu does! % % -------------------------------------------------------------------- #042 Example: Flood waves in rivers. Viscosity solution cannot be justified physically. There is no analog of the ``look ahead'' preventive driving of Traffic flow. Fluid particles keep on going till catastrophe strikes: the shock layer structure involves turbulent dissipation, etc. No simple 1D model possible. Observation indicates that thin transitions occur --- so theory applies. % % -------------------------------------------------------------------- #043 Example: Numerical viscosity. Even if "non-physical", the addition of viscosity (in conservative form) to the equations, when shocks are known to occur, prevents wave breaking and gives structures that (macroscopically) behave correctly. Hence, one can use this to stabilize numerical schemes. % % -------------------------------------------------------------------- #044 First order quasi-linear 1-D systems of equations. Classification. Hyperbolic systems and characteristics. Domains of dependence and influence. Examples. First order quasilinear systems of equations u_t + A*u_x = F(x, t, u), where A = A(u, x, t). Characteristics and characteristic form of the equations. Example: linear, constant coefficients, no sources, case. Hyperbolic if A is real diagonalizable. Example: general solution for a hyperbolic system where A is constant and F = 0. In general, F \neq 0, characteristics couple. Domains of dependence and influence. Examples: Linear Gas Dynamics (acoustics). Sound waves, general solution. Wave equation. Reduce to form above. Klein Gordon equation. Characteristic form. Domains of dependence and influence. % % -------------------------------------------------------------------- #045 Examples of first order 1-D hyperbolic systems. Linear acoustics. Example: Linear acoustics in 1-D. Exact solution by characteristics. System equivalent to wave equation. % % -------------------------------------------------------------------- #046 Systems of the form Y_t + A*Y_x = 0. Show ill-posed if complex eigenvalue exists. Motivation and physical meaning: When A is not a constant (particularly if A = A(u)) the system can change type, switching from well posed to ill posed. Example: stratified flows. --- Interpretation of the Richardson # criteria as a change of type (not an instability issue, as often portrayed). --- Physical reason: mixing occurs, stratification assumption fails. Ad-hoc fix: add vertical diffusivity in mass equation (mixing). % % -------------------------------------------------------------------- #047 Example: Wave equation. Solution of the initial value problem: D'Alembert. Domains of dependence and influence (Note that they involve the full wedge, not just the characteristics at the edge, for data u and u_t). % % -------------------------------------------------------------------- #048 Y_t + A(Y)*Y_x = 0; hyperbolic [A real diagonalizable] Simple waves (from simple eigenvalues and eigenfunctions). Wave breaking occurs. As in the scalar case, characteristics cross. Note simple waves also provide rarefaction wave solutions. % % -------------------------------------------------------------------- #049 Breakdown of solutions: need to input appropriate physics. An example is when shocks apply. Shocks and shock conditions for systems of conservation laws. Rankine Hugoniot conditions. Derivation of the Lax entropy conditions as needed for causality. % % -------------------------------------------------------------------- #050 Example: Gas Dynamics, use mass-Lagrangian coordinates v_t - u_z = 0 and u_t + p_z = 0; p = p(v) convex with dp/dv = - rho^2 a^2 < 0. % % -------------------------------------------------------------------- #051 EXTRA DETAILS for Gas Dynamics in 1-D (Isentropic). Formulation in terms of mass Lagrangian coordinates. Riemann Invariants and simple waves. Wave breaking. Shock conditions (Rankine-Hugoniot ) for systems. Lax entropy: explain how it works for causality. Shocks in the p-v plane. Right and left shocks. Lax entropy equivalent to compressive shocks. Shock curve: for a fixed ``right'' state on a ``right'' shock, states in phase space (u, v) that can be reached by a shock. Similar curve exists for left shocks, starting from left state. Rarefaction curve: Same idea s for the shock curve. Write rarefactions using characteristic form, in particular: Riemann Invariants. RIEMANN PROBLEM: Show how to solve using the shock/rarefaction curves as a coordinate system in phase space. Describe solution in space-time. General systems: there are N shock curves and N rarefaction curves. At least locally they can be used to solve the Riemann problem. In general not always clear as the states on the right and left in a Riemann problem get further apart. % % ========================================================================== % EOF