18.306-MIT. Advanced PDE with Applications. Spring 2013. Tu and Th 9:30--11:00. Room 2-139 Rodolfo R. Rosales. Lecture Summaries. [PSQ] = Problem Set Question. 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. % ========================================================================== % --------------------------------------------------- Lecture #01 Tue-Feb-05 TOPICS: Mechanics of the course. Example pde. Initial and boundary value problems. Well and ill-posed problems. Introduction: Syllabus issues; grading; problem sets; exams; lecturer; MatLab; WEB page; etc. News and Updates link in WEB PAGE % % -------------------------------------------------------------------- #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 C-K theorem, and need very strong restrictions (e.g.: analytic functions.) % % -------------------------------------------------------------------- #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) = 0, 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 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. % % ========================================================================== % --------------------------------------------------- Lecture #02 Thu-Feb-07 % TOPICS: Well and ill-posed problems. Conservation laws and pde. Integral and differential forms. Closure strategies. Quasi-equillibrium. Examples. % % -------------------------------------------------------------------- #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 ask: well, but these are silly examples. On this, 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 that are known to be ill-posed under some conditions, but it is not known why or how to fix them. Something is wrong and/or missing from the models. Examples in: multi-phase flows, continuum models of phase transitions, square wave model for detonation waves. (C) Possible Fix: filtering --- get rid of the high frequencies. Works if filtering makes sense within context of problem. 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 \ @#@ SKIP for now. with interface conditions. / May see later. It can also lead to nonsense if applied mechanically. (D) IMPORTANT. Difference between ill posed with arbitrarily large growth rates and sensitivity to initial conditions in chaotic dynamics. % % -------------------------------------------------------------------- #006 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. % % ========================================================================== % --------------------------------------------------- Lecture #03 Tue-Feb-12 03a Review: Well and ill-posed problems. Finish #005: points B and D. 03b Review: Conservation laws and pde. Integral and differential forms. 03c Do derivation of the string equation, as a further example of the method of conservation laws to get pde. 03d Do derivation of the traffic flow equations. Then consider the tunnel problem, and flow control by doing something at the ends [example of where causality plays a role]. See item #007. 03e Closure strategies. Quasi-equillibrium. Relationship with thermodynamics. See items #008 and #009. % % -------------------------------------------------------------------- #007 A simple example of well/ill posed problem. 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). In these cases, the problem is well posed: the solution exists, is unique, and depends continuously on data. Else, no solution that does not break causality exists. % % -------------------------------------------------------------------- #008 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). 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. 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. EXAMPLE: Euler equations of gas dynamics (1-D) and closure via equilibrium thermodynamics. EXAMPLE: Elasticity, will come back to this later. EXAMPLE: heat equation for a rod. Conservation of heat and Fick's law. Derivation of equation in more than 1-D. Boundary conditions: Dirichlet, Neuman, and Robin/mixed. Other pde we will study: Wave, Laplace, Poisson, Helmholtz, Eikonal, Euler equations of Gas Dynamics, Navier Stokes equations, etc. Will derive as they occur. % % ------------------------------------------------------------------- #009a Conservation laws: HIGHER ORDER EFFECTS. Transport/Dissipative terms, dispersion, etc. Eqn. of state depends on derivatives of the densities. Examples: Heat flow. In general: diffusion (write/derive diffusion eqn.). Preventive driving in Traffic Flow. Navier-Stokes (corrections to Euler equation of state). Burgers' equation. Important in limiting the size of the gradients that can occur, and preventing infinities from developing. Will talk more about this later. % % ------------------------------------------------------------------- #009b % SKIP [for now, at least]. % 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 = O(tau2), only the E1 modes can equillibrate. Model this by: d(e-E1)/dt = -(1/tau)*(e-E1-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 [for now, at least]. % Dispersion Loosely: Time evolution of wave phenomena where the wave speed depends on the wavelength [hence 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 equation: 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 solution: 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. Namely: (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). % % ========================================================================== % --------------------------------------------------- Lecture #04 Thu-Feb-14 04a Do #006. 04b Do Point D in #005 04c Microscopic source of causality: higher order effects (e.g. diffusion). We already saw this for the heat equation. 04d Do item #007, or leave for a problem set. 04e Closure strategies. Quasi-equillibrium. Relationship with thermodynamics. See items #008 and #009 04f Do #010 % % -------------------------------------------------------------------- #010 Higher order (diffusion) in the LWR-Model of Traffic Flow. Go back to #007 and clarify the effects of neglected effects. Causality. 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, which it is possible for some problems --- however, keep in mind that, even when "possible" (if the model is only approximate, as most models are) the "small" corrections may limit how much of this can actually be done. % % ========================================================================== % --------------------------------------------------- Lecture #05 Thu-Feb-21 % % -------------------------------------------------------------------- #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. % =============================================================== % % =============================================================== % % We now will start following, more-or-less, the book by Salsa. % % =============================================================== % % =============================================================== % % -------------------------------------------------------------------- #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. 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 Construct solution to #S03b using separation of variables. Process works and produces a solution as long as g is L^2. Note solution is smooth for t > 0, even if initial data is not. Solution achieves initial data in L^2 sense for g in L^2. If g is better than L^2, then initial data achieved in better senses. Note: above presumes knowledge of Fourier series. REVIEW if needed. Process can be generalized to other B.C. ................... Examples. Robin requires some knowledge of Sturm Liouville problem. REVIEW if needed. % % ========================================================================== % --------------------------------------------------- Lecture #06 Tue-Feb-26 % % -------------------------------------------------------------------- #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 that A is self adjoint & negative definite. Actually, only need self adjoint and bounded from below. % % -------------------------------------------------------------------- #S06 Maximum principles (heat equation). Intuitive/physical reasoning: heat flow away from any local maximum and into any local minimum. Proof: First for trivial case u_t - \Delta u < 0, or u_t - \Delta u > 0. Extend to u_t - \Delta u <= 0, or u_t - \Delta u => 0. Quote strong max. principle. % % -------------------------------------------------------------------- #S07 Review of fundamental solutions/Green functions for ODE. Elementary theory of distributions, weak derivatives, delta function, Heaviside function and derivative. Calculate delta(alpha*x) in 1-D using the definition. % % -------------------------------------------------------------------- #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. % % ========================================================================== % --------------------------------------------------- Lecture #07 Thu-Feb-28 Finish from last lecture: #S05 Sturm Liouville, Self-Adjoint, Connection with ode. Normal operators. Numerical stability computing eigenvalues. Analyzing stability of solutions. Spectral methods. #S08 Review dimensional argument, as well as similarity argument. % % -------------------------------------------------------------------- #S09 Fundamental solution, n > 1. Derivation by similarity arguments, same as for #S08. Final formula involves "area" of unit sphere S_{n-1}. Derivation using tensorial nature of n-dim heat equation. Get formula for "area" of unit sphere S_{n-1}, any n. Again, as t \to 0, get delta in n-D. Solution initial value problem. Generally smooth for any t > 0. % % -------------------------------------------------------------------- #S10 Examples of fundamental solutions for special bounded and semi-bounded domains, with various BC [Dirichlet, Neumann, Periodic]. Exploit symmetry: Reflection principle. % % ========================================================================== % --------------------------------------------------- Lecture #08 Tue-Mar-05 Recap dimensional argument [heat within region of size sqrt{D*t}]. Do #S10 % % ========================================================================== % --------------------------------------------------- Lecture #09 Thu-Mar-07 % % -------------------------------------------------------------------- #S11 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 ... note that fundamental theorem of calculus preserved. % % ========================================================================== % --------------------------------------------------- Lecture #10 Tue-Mar-12 Recap use of symmetries to get Green functions in domains with boundaries. Formula for area of sphere using heat equation. [#1] ---------------- #S12 Do Random walks and Brownian motion in 1-D .......................... [S13] Do Critical Mass for Fission. [#2] ---------------- #S14 [#1] From 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)." [#2] From problem series "Diffusion and probability" ["Critical mass"]. Also Salsa's book. % % ========================================================================== % --------------------------------------------------- Lecture #11 Thu-Mar-14 Finish Random Walks/Brownian motion. --- Formulas for expectation. --- Justification using Fourier transforms and normal modes. Green function in multi-D and methods of images. -------------------- # S15 --- Do generic case. Images NEED NOT be deltas. Examples with Robin boundary conditions: See: Green's functions #04 [IVP, heat equation in the semi-infinite line]. See: Supplementary material, methods of images, Robin BC in an interval. Both in the problem series: Point sources and Green's functions. Do Duhamel's principle .............................................. [S16] See Salsa's book: 2.8.3 pp. 71-73. Do Tychonov's example and global maximum principle .................. [S17] See Salsa's book: 2.8.4 pp. 74-76. Skip 2.9 pp. 77-89. % % ========================================================================== % --------------------------------------------------- Lecture #12 Tue-Mar-19 Finish with topics in lecture #11. Tychonov's example and Duhamel's principle. In particular: give more details about Tychonov's example. % % ========================================================================== % --------------------------------------------------- Lecture #13 Thu-Mar-21 Still with Duhamel and Tychonov. Nonlinear diffusion equation and finite propagation speed. ---------- [S18] The porous media equation. See also Salsa 2.10.1 pp 90-92. Reaction-Diffusion equations. ---------------------------------------- #S19 Fisher's equation. Salsa 2.10.2 pp 93-96. SKIP Briefly mention: Waves and patterns in reaction diffusion equations. Spiral, bulls-eyes, etc. % % ========================================================================== % ----------------------------------------------- SPRING VACATION Tue-Mar-26 % ========================================================================== % ----------------------------------------------- SPRING VACATION Thu-Mar-28 % % ========================================================================== % --------------------------------------------------- Lecture #14 Tue-Apr-02 Boundary Layers. Example u_t = (epsilon*u_x + u)_x for x > 0. 0 = epsilon*u_x + u at x = 0. Motivation for the problem. Solution for 0 < epsilon << 1 and 0 < t << 1/epsilon. FROM: "Various lecture notes for 306", "Advection-diffusion in 1-D". % % -------------------------------------------------------------------- #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: 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. % % ========================================================================== % --------------------------------------------------- Lecture #15 Thu-Apr-04 Finish with boundary layers [screwed it up in lecture 14]. % % ========================================================================== % --------------------------------------------------- Lecture #16 Tue-Apr-09 % % -------------------------------------------------------------------- #S21 From: "Various lecture notes for 18306" Section: Laplace and Poisson equations - harmonic functions. DO 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]. % % ========================================================================== % --------------------------------------------------- Lecture #17 Thu-Apr-11 Finish with #S21. % % -------------------------------------------------------------------- #S22 From: "Various lecture notes for 18306" Section: Laplace and Poisson equations - harmonic functions. DO Subsection: Poisson's formula and Harnack's inequality. DO Subsection: The fundamental solutions. [A] [B] [A] 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 the Green's functions with only a few terms, at least for moderate times]. 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 be used to accelerate the convergence. e.g.: f(x) = sum_{n \neq 0} (1/|x-n|-1/|n|-|x|/n^2) + |x| sum (1/n^2), 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] Did not cover applications of the fundamental solution, when combined with Gauss/Green's theorems, to solutions of the Poisson equation in a domain \Omega, with boundary \partial \Omega. e.g.: single and double layer potentials, etc. Will return to this, if time allows, after doing some hyperbolic stuff. % % ========================================================================== % -------------------------------------------------- PATRIOTS DAY Tue-Apr-16 % % ========================================================================== % --------------------------------------------------- Lecture #18 Thu-Apr-18 We now start with hyperbolic equations. We do not follow Salsa anymore here. Definition of hyperbolic and elliptic for scalar second order pde in 2-D using the symbol. Example: wave equation and Laplace equation. TOPICS: First order scalar pde. Characteristics. Examples of solutions by characteristics. Domains of definition, influence, and dependence. You can find this in any pde book. Start with the case: Equation is for a scalar unknown, linear, in two independent variables (x, t) or (x, y), with data given on some curve. % -------------------------------------------------------------------- #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. 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 physical system modeled by the equation. General formalism: Write solution in characteristic coordinates x = x(s, t) \ s = parameter along data curve y = y(s, t) | t = parameter along charact. u = u(s, t) / Then invert: s = s(x, y), t = t(x, y) to obtain the solution. % % -------------------------------------------------------------------- #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. [#] 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]. [#] "Only once" is needed because the solution is fully determined, along each characteristic, by a single value. [#] "Transversally" is 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. --- 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. % % ========================================================================== % --------------------------------------------------- Lecture #19 Tue-Apr-23 TOPICS and IMPORTANT CONCEPTS: Characteristics as curves where pde reduces to ode, at least in 2-D Data can be given on some curve. Domain of influence of data. Domain of dependence and CFL. Case (x, t) causality for data on boundaries. Allowed boundary conditions. Semilinear case: possible blow up of solutions along charact. Begin with quasilinear case. --- Graphical interpretation of solution by characteristics. --- Conservation. Wave steepening and breaking. % % -------------------------------------------------------------------- #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 meetting 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 shapped 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)). 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 brakdown 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]. % % ========================================================================== % --------------------------------------------------- Lecture #20 Thu-Apr-25 % =========================================== % % *** REMEMBER: DROP DATE is Mon. Apr. 29 *** % % =========================================== % TOPICS: Quasilinear Equations. Back to the physics. Integral form of the conservation laws. Shocks. Examples from traffic flow. Explain shock laws, and entropy. Why is entropy needed. See #029 part A. More later. First peek at entropy for equations with two sets of characteristics. % % -------------------------------------------------------------------- #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. Note that the path in space time of the last car through % the light is right in the zone where characteristics % cross, and that the characteristics on each side converge % 024 A into it and ``die'' there. This is how the crossing of the % 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. Again: observed behavior has a discontinuity, but now the % speed at which the discontinuity moves is not as % 024 B 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] % 025 A Derive also by looking at conservation in shock frame Flux in = (u_-s)*rho_ \ equality leads to Flux out = (u+-s)*rho+ / 025 A Apply 025 A to the discontinuity in 024 B, then again show that the analog of 024 A applies: The path of the discontinuity is inside the region where % the characteristics cross, with the characteristics on % 025 B each side converging into it and ending there. Thus the % crossing is avoided. % Note that 025 A reduces to the characteristic speed for infinitesimal discontinuities. Note graphical interpretation of 025 A: slope of secant line connecting shock states in rho-q diagram. % % ========================================================================== % --------------------------------------------------- Lecture #21 Tue-Apr-30 TOPICS: Shocks in Kinematic waves (Traffic Flow, River Flow, etc). Conservation and Rankine Hugoniot Jump Conditions. Entropy conditions, irreversibility. % % -------------------------------------------------------------------- #026 In #024 and #025 we saw examples where discontinuities (*) (*) Of course: 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. Such discontinuous transitions occur regularly both in Traffic Flow and in River Flow. Thus 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) will be needed for a good theory. We will get to them later. Note 2: the fact that, for sufficiently slow rates of change, 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 will generalize this theory for systems of conservation laws (e.g.: Euler equations of Gas Dynamics). A) The p.d.e. (and the solution by characteristics) applies in regions where the solution is smooth enough (say: continuous partial derivatives) B) Simple discontinuities 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 will show how to generalize this to solutions that are much less smooth than this. C) Across each discontinuity the Rankine Hugoniot jump conditions apply: dS/dt = [q]/[rho]. This follows from conservation, and guarantees it. Note that this reduces to the characteristic speed for infinitesimal discontinuities. Note 4: 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. D) PROCESS: Shocks are introduced to avoid the 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. % IMPORTANT: for this to be possible, and to obtain a well posed model, % a RESTRICTION ON THE ALLOWED DISCONTINUITIES (shocks), IN % ADDITION to the RANKINE-HUGONIOT CONDITIONS, IS NEEDED. This is the % LAX ENTROPY CONDITION. % Conservation alone not enough: allows unphysical solutions. See #028 % The ENTROPY CONDITION is related to the fact that: % Shocks introduce causality (arrow of time) and irreversibility % into the equation. Solutions that do not have shocks can be run % backwards in time (at least till the characteristics cross in the % past), but solutions with shocks cannot. We get back to this in detail later (see #029-030-031), but first some examples. % % -------------------------------------------------------------------- #027 % ................. To lecture #022 RAREFACTION FANS. Lessons from example: Traffic flow, red light turns green. This 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: A) 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. B) 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. C) 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. ............................ partly covered in lecture #20. A) 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''. B) Well posed argument. -- Inspect the various scenarios for characteristic patterns near a shock, and 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): This leads to two problems (one on each side of the shock), each fully determined by the initial data, plus one equation that then determines the shock path. C) High order effects argument. Zero viscosity limit; shocks as internal layers. -- Assume a more detailed model, incorporating the physical effects that become important in the shock region. Then the shocks should occur in the more detailed model solutions, in the limit where the high order effects vanish. -- Example: zero viscosity limit. When the higher order effect is a diffusion/viscosity: such effects cause a flow against the gradient (Fick's law), which fight steepening and 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 types of high order effect do NOT produce shocks. Examples: zero dispersion limit, ``collisionless'' shocks in plasmas. For a shock you need a sharp, thin, transition region. If the formation of steep gradients triggers waves that radiate away from the breaking region: cannot use shocks. D) 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 drawings: 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 "die" 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 and entropy conditions as above. 1) Argue that the "information" contents in the graph of a function rho = rho(x, t) is given by the ``wiggliness'' it has. For a given fixed area, the more variation in the function, the more information. Examples: -- Constant: just one number characterizes it. -- Sinusoidal. Need mean, amplitude, frequency and phase. 2) How to "measure" the information? Propose (1/2)*rho^2 as "information density". As the wiggles increase, for a given area, the integral of (1/2)*rho^2 goes up. More generally, any strictly convex function f(rho) can serve as a measure of information density. 3) 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 now 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 that information is lost iff and only iff the Lax Entropy conditions apply. Do the calculations for the case Q(rho) = (1/2)*rho^2. Leave the general case for a problem set. -- Connection with high order effects and viscosity: Will show later Lax Entropy condition is equivalent to the equation with shocks being the limit as nu --> 0 for rho_t + q_x = nu*u_xx, with nu > 0. For this equation, easy to see that d/dt (int rho^2) < 0. % % ========================================================================== % --------------------------------------------------- Lecture #22 Thu-May-02 Do example from #027 Kinematic equations with a non-convex flux ... #032 Shocks for equations with source terms ....... #032 Riemann problems and numerical solutions ..... #034, #035, #036, #37. 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 Godunov-type methods. Example: SKIP Riemann problem for kinematic wave eqn. with convex/concave flux. Example: [PSQ] Riemann problem for a conservation law with a point source term. Example: SKIP [PSQ] Riemann problem for kinematic wave eqn. with a flux that is not concave/convex [as in #32]. % % -------------------------------------------------------------------- #032 Kinematic equations with a non-convex flux 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] 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. % % ========================================================================== % --------------------------------------------------- Lecture #23 Tue-May-07 Shocks and weak solutions. Weak formulation of the equations, including initial conditions. 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. Characteristics as locus of possible weak singularities in the solution. Connection with the "symbol" of the equation: write equations in (zeta, eta) coordinate system, where zeta = constant are possible singularity lines. Elliptic as equations that do not accept any singularities. % % -------------------------------------------------------------------- #033 Shock jump conditions and generalized derivatives. Explain notion of generalized derivative. Examples: Heaviside function, delta function, principal value, etc. Use it to get shock jump conditions by matching ``singular'' part in conservation form of the equation. Note that product rule does not work for generalized derivatives, so using the correct conservation form is crucial. % % -------------------------------------------------------------------- #034 Riemann problems and numerical solutions. --- Introduce Riemann Problem. %>> See examples #035/036. <<% --- Whole problem is encoded into the Riemann Problem, including the R.K. jump conditions and the Entropy cond. If you can do the R.P., then you have, in principle, everything. --- Godunov's type methods. --- Higher order and generalized Riemann problems. van Leer. --- Higher order and semi-discrete, using only Riemann problem. % % -------------------------------------------------------------------- #035 % SKIP % Riemann problem for kinematic wave eqn with convex/concave flux. Riemann problem for: u_t + Q(rho)_x = 0 -- Case Traffic Flow Q concave -- Case River Flows Q convex % % -------------------------------------------------------------------- #036 % SKIP [PSQ] % Riemann problem for a conservation law with a point source term. Example: 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 the origin implies there is a discontinuity there, and appropriate jump conditions must be given, restricted by causality. % % -------------------------------------------------------------------- #037 % SKIP [PSQ] % Rankine Hugoniot and entropy conditions for u_t + (0.5*u^2)_x = delta(x). 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. Use this approach 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. % % -------------------------------------------------------------------- #038 % SKIP % General description of how to do Riemann problems (preview of what happens for systems). Shock and rarefaction branches to-and-from a point in state space. Then connect states on the right and left going through intermediate states Examples: u_t + q_x = 0 for q = q(u) cubic. The examples done for #037 Preview of systems: what happens for systems with two variables. % % -------------------------------------------------------------------- #039 % THIS WAS COVERED EARLIER. % Shock structure produced by more detailed physics: Viscosity. Assume that the higher order effect is diffusion/viscosity. As explained earlier, such effects cause 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. In short, generally expect that -- Diffusion fights steepening: induces extra flow down the gradient. -- Effects of the large gradients remains local (only where the wave steepening happens). -- Steeping eventually stopped and a local, thin, transition develops ==> discontinuity in limit. % % -------------------------------------------------------------------- #040 % PARTIALLY COVERED EARLIER. SKIP THE REST. % % 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 NON-CONVEX FLOWS Examine (graphically) the PROBLEMS that ARISE when Q is neither convex, nor concave. Show that a ``shock'' solution may not exist. Show: 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 % SKIP. [PSQ]. % .............. This is related to #031. % Shocks, dissipation, and "information" contents. (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 % SKIP % 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: shock layer structure involves turbulent dissipation etc. No simple 1-D model possible. Observation, though, indicates that thin transitions occur, so theory applies. % % -------------------------------------------------------------------- #043 % SKIP. % 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. % % ========================================================================== % --------------------------------------------------- Lecture #24 Thu-May-09 Existence of nontrivial (local, infinitesimal) high frequency solutions and characteristics. Hyperbolic: can write "everything" as linear combinations of these high frequency solutions. DO #044. Get characteristics first by using weak singularities to get eigen-equation and symbol (determinant of eigen-equation). DO #045-#046 DO #042 and #043 is time permits. % % -------------------------------------------------------------------- #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 dependece and influence. % % -------------------------------------------------------------------- #045 Examples of first order 1-D hypebolic 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). % % ========================================================================== % --------------------------------------------------- Lecture #25 Tue-May-14 #25.0 Finish with #046 #25.1 Example of hyperbolic system in 1-D. p-system [Gas dynamics, shallow water, etc]. Eulerian and Lagrangian formulation. Characteristics. Riemann invariants form and simple waves. Rankine Hugoniot jump conditions. Rayleigh line and p-v diagram. Entropy condition. #25.2 Simple waves for general hyperbolic system % % ========================================================================== % --------------------------------------------------- Lecture #26 Thu-May-16 #26.1 Hyperbolic in n-D. #26.2 Characteristics as high frequency fronts and generalized Eikonal. #26.3 Eikonal for wave equation and bi-characteristics. % % ========================================================================== % -------------------------------------------- EXTRA TOPICS/FURTHER DETAILS. % % -------------------------------------------------------------------- #047 Example: Wave equation. Solution of the initial value problem. D'Alembert solution. Domains of dependence and influence. Note: full wedge for data u and u_t % % -------------------------------------------------------------------- #048 Y_t + A(u)*Y_x = 0. Simple waves (from 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 appart. % % ========================================================================== % %% EOF