LECTURES SCHEDULE FOR 18.306. Spring semester 2008. Lect. # & date =========================================================== General mechanics of class: grading, problem sets, #1 Wed-Feb-06 exams, class and e-mail list, MatLab, WEB page, etc. % ========================================================= Some facts about ode's: Standard form of n-th order ode [dY/dt = F(Y, t)]. Quote general existence, uniqueness, etc. theorem for ode IVP (IVP is well posed ... define term). 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. Examples, show: some have a unique solution, others have no solution, others have many (even infinite). See d^2u/dx^2 + m^2*u = 0 on 0 < x < 1, with u = 0 at end points (no solution or infinitely many). Completeness of IVP ode theory reflected in numerics: IVP solvers solid and robust. BVP solvers less so. % ========================================================= Simple facts about pde's: Define what pde is. There is no standard form, though, nor any general well-possedness theorem. PDE solutions include free functions, not constants. Simple example: u_x = 0. Solve with data on curves: which curves does this work for. Characteristics as curves on which pde reduces to ode. Simplest pde in two variables: u_t + c*u_x = a*u. Find general solution by characteristics. Show which BV problems make sense. Show well posed: solution exists, is unique, and depends continuously on data. Physical meaning/importance of being well posed. Other pde we will study: wave, Laplace, Poisson, heat. Derive heat equation for a rod. Use conservation of heat and Fick's law. Boundary conditions: Dirichlet, Neuman, & Robin/mixed. % ========================================================= Examples of ill-posed: IVP for backward heat equation. #2 Mon-Feb-11 Solutions to u_t + u_xx = 0 arbitrarily large for any finite time t > 0, with arbitrarily small IV. IVP for Laplace (do the same). Examples from applications: some fluid models with phase transition models. Important difference between illposed with arbitrarily large growth rates and sensitivity to initial cond. in chaotic dynamics. % ========================================================= Conservation Laws: Continuum limit. Densities, fluxes and sources. Integral and differential form of a conservation law in one and more dimensions. Closure, quasi-equillibrium and thermodynamics. Typically, gives rise to hyperbolic equations. Examples: kinematic waves (Traffic Flow & River Waves), Differences in Q = Q(rho) for these two last cases, Euler equations of Gas Dynamics. Elasticity. #3 Wed-Feb-13 Higher order effects (transport). Dissipative terms. Examples: Burgers equation; Navier Stokes; etc. Important in limiting size of gradients and preventing infinities from developping. Relaxation effects (ionization in gases): d(e-E1)/dt = -(1/tau)*(e-E1-E2) d/dt = material derivative. E1 = equill. energy in faster degrees of freedom. E2 = equill. energy in slower degrees of freedom. tau = equillibration time. Drivers response time in traffic flow dq/dt = -(1/tau)(q - Q(rho) + nu*rho_x) % ========================================================= Classification of pde's: Linear, semilinear and quasilinear. Examples for 1st order, scalar, in two indep. var. Hyperbolic, elliptic, parabolic, dispersive. Hyperbolic ~ waves with finite number of speeds Dispersive ~ waves with frequency dependent speed. Elliptic ~ equillibrium problems. Parabolic ~ transport effects. Dispersive Fourier series solutions of constant coefficient linear evolution equations and explanation of dispersive -- breakdown of coherence needed for localization if d\omega/dk is not a constant for dispersion relation omega = omega(k). Derivation of the linear string equation and of the string over an elastic bed equation. % ====================================================== Students expected to be familiar with these concepts for linear waves: 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 = F(k). Phase and group speed. --- Examples: KdV, water waves, NLSchodinger, etc. --- The wave equation. % ====================================================== Generalize to nonlinear problems: Nonlinear dispersive wave theory; u = F(theta, A). % ========================================================= #4 Mon-Feb-18 Recap types of pde's: --- Linear, semilinear, quasilinear, the rest. Classification of pde's (made precise later). --- Hyperbolic: Typically describe a time evolution. They involve wave phenomena, with a discrete set of velocities. Time reversible. Signals have a finite speed of propagation; localized data remains localized. --- Elliptic: Typically describe equillibrium situations. Ill posed for problems involving time. --- Parabolic: Typically describe a time evolution involving transport effects (diffusion, viscosity, etc.) No bound on signal speed (infinite). Not time reversible. This list is not inclusive. % ========================================================= Dispersive pde's: describe time evolution with wave phenomena where the wave speed depends on the wavelength [hence continuous of possible speeds]. A dispersive pde can be hyperbolic, or not. Example 1: u_tt - u_xx + u = 0. solutions u = Re{A*exp[i*(k*x-omega*t)]} where omega^2 = 1 + k^2 This example is also hyperbolic Dispersion condition: omega = F(k) is real valued and d^2 omega/dk^2 does not vanish. Localized initial conditions loose phase coherence and de-localize (disperse). Example 2: u_tt - u_xx + V'(u) = 0, where V is convex. Generalizes the 1st example (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). Equation for U can be show to be 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 the mass can be adjusted to make them 2*pi periodic. Hence, get omega^2 - k^2 = m(E). In a nonlinear dispersive system, the wave number is a function of the wave number AND the wave amplitude. % ========================================================= General solution of scalar, 1st order, quasilinear pde by CHARACTERISTICS (you can find this in any pde book). Start with 2 independent variables: (x, t) or (x, y), with data given on some curve. 1st 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) 2nd Linear, variable coefficients #5 Wed-Feb-20 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 what happens when singularities arise in the coefficients. In particular: discontinuities can 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. Example: x*u_x + y*u_y = 0. Show general solution for y > 0 is u = F(x/y). In polar coordinates: u = f(theta). Problems: u = e^x on y = x^2. Solution defined for y > 0. u = e^x on x = 1. Where is solution defined? General formalism: 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) / Invert: s = s(x, y), t = t(x, y) and obtain solution 3rd Semilinear. Example: u_t + u_x = u^2 and u(x, 0) = f(x). where is solution defined? Issue: when singularities arise, math. goes "bad" and modeling hypothesis have to be revised. Crap example: if the equation 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 phenomena of self-focusing yields singularities. 4th Quasilinear. NEXT LECTURE. CONCEPTS: Characteristics as curves where pde reduces to ode. Data can be given on some curve. Domain of influence of data. Case (x, t) causality for data on boundaries. Semilinear case: possible blow up of solutions along charact. % ========================================================= Quasilinear case: crossing of characteristics. #06 Mon-Feb-25 Point out: in linear and semilinear case, characteristics known a-priori, and provide "good" curvilinear coord. Quasi-linear: they depend on solution. Example: u_t + c(u)*u_x, u(x, 0) = f(x). -- Solution by characteristics. x=c(f(s))*t+s, u=f(s). -- Implicit form of solution. u = f(x-c(u)*t). -- Point out characteristics cross if (d/ds) c(f(s)) < 0. Drawing and formula x_s = 1 + t*f'(s)*c'(f(s)). First time of crossing. -- Graphic interpretation and wave steepening & breaking. -- Break-down of mathematical model. What happens after wave breaking? Must go back to physics. -- Assume neglected effects: 1) fight steepening by inducing extra flow down the gradient. 2) effects of large gradients remains local. If so, wave steeping eventually stopped and a local, thin, transition occurs ==> discontinuity in limit. SHOCKS. Examples: traffic flow and river flows. Augmented model: -- p.d.e. applies in smooth regions. -- simple discontinuities allowed along curves x = S(t). -- Need conditions across discontinuities to solve the p.d.e on each side (signaling problems) + determine shock path. HOW many conditions do we need? Will check this later. -- Note: must make sure augmented model is well posed. -- Conservation must hold across discontinuities! Can use integral form to obtain condition: Jump condition: dS/dt = [Q]/[rho] Yields Characteristic speed as [rho] --> 0. Note: more than one integral form can hold for the "same" #07 Wed-Feb-27 equation. Must select physical one to get correct jump conditions. Example: u_t + (1/2*u^2)_x = 0 and (1/2*u^)_t + (1/3*u^3)_x = 0 ===> same p.d.e. Example: Traffic flow, red light turns green. -- Disc. initial data. -- Note a solution with a discontinuity allowed, but NOT stable. -- Get expansion fan solution by "smearing" discontinuity. -- Argue: not only is the "correct" mathematical solution, but it is also the one observed in physical situation. -- Do case of Q quadratic. This example points to two issues a) How to deal with discontinuities in the data that yield "gaps" in the characteristic field: smear data at discontinuity [get back to this later]. b) We MUST IMPOSE RESTRICTIONS ON WHAT DISCONTINUITIES ARE ALLOWED. Conservation alone not enough, allows unphysical solutions. Leads to ill-posed problem. THIS IS THE NEXT TASK! % ========================================================= ENTROPY CONDITIONS AT SHOCKS. TIME ARROW. IRREVERSIBILITY. Formation argument: when do we need shocks? When does wave breaking occur? Characteristics must cross/converge. c- > s > c+ <--- "Lax entropy condition." Well posed argument: inspect that various scenarios of characteristic patterns near shock, and see which one leads to well posed problems on each side of the shock, plus determination of the shock path. c_- > s > c_+ leads to two problems (one on each side of the shock) fully determined by the initial data, plus one equation that then leads to determination of shock path. This all works out provided jump condition is consistent with c- > s > c+. Investigate consistency: 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: can of worms. Not clear what to do in general. Will come back to this later. Show some drawings ... problem is that Q need not stay on only one side of the secant line. High order effects argument: look for traveling wave #08 Mon-Mar-03 solutions of equation with higher order effects. Start with rho_t + c(rho)*rho_x = nu*rho_xx (assume non-dim, with nu small). Then argue that shock transition must have width nu (scale t/nu and x/nu) for higher order effect to balance nonlinear steepening. Argue that this (locally) makes shock path into a straight line, and the shock profile into a travelling wave. Examine solutions of the form u = F((x-s*t)/nu). When do they exist and provide a smooth connection across discontinuity in nu --> 0 limit? Examine O.D.E. problem for F {a boundary value problem}. When does solution exist? Use convexity and/or concavity of Q = Q(rho) to get complete theory in this case. Get same conclusions as before: jump conditions and characteristics converge on shock path. Examine the problems that arise when Q is neither convex, nor concave (graphically). All along: zero in the graphical interpretation of the shock and entropy conditions in the rho-q plane, and the role that convexity/concavity plays. Examples: --- Solve red-light turns green turns red type of problem for equation u_t + (u^2/2)_x = 0, with initial data u = 1 for x > 0 and u = -1 for x < 0. --- Solve signaling problem for u_t + (u^2/2)_x. --- Solve u_t + q_x = 0, with q = -(1/2)*u^2 + (1/4)*u^4 (non convex) for initial data u(x, 0) = alpha*sign(x). --- alpha > sqrt(2) yields steady state shock. --- 0 < alpha < \sqrt(2) yields two shock waves connected by a rarefaction. #09 Wed-Mar-05 REASON FOR THE NAME ["Lax entropy condition"]. Recall "quasi-equillibrium" is thermodynamics in gas dynamics. There analog condition equivalent to 2nd law of thermodynamics: entropy must not decrease as fluid particles cross shock layer. Get to this later. Irreversivility of augmented system. Once shocks form, cannot go back in time. Characteristics "die" at shocks, and the info they carry is LOST. Second law of thermo. Shocks and dissipation: 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. 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. 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. Consider Burgers' equation u_t + u*u_x = nu*u_xx and | calculate equation for the evolution of the information | Not done. Do contents int (1/2)*u^2. Show it decreases due to nu*u_xx | it next lect. term. Plug in shock layer solution and show that the | amount of information loss DOES NOT go to zero as nu does! | % ========================================================= Envelope of characteristics and shock formation/locus. Introduce notion of envelope and derive equations for it: Family of curves given by F(s, x, y) = 0. Then envelope given by F(s, x, y) = F_s(s, x, y) = 0. Show that the two definitions of envelope lead to the same answer. Do example: F = x*cos s + y*sin s - 1. #10 Mon-Mar-10 For the equation u_t + c(u)*u_x = 0, u(x, 0) = U(x) smooth, show that the envelope of the characteristics corresponds to the curves in space-time along which the derivatives of the solution become infinity. These curves delimit region where solution is multiple valued. Shock is inserted somewere in there, and starts at the cusp of the envelope. % ========================================================= Examples of a case where shocks are the WRONG answer to the multiple values: --- Dispersive waves modulation. --- Small dispersion limit for KdV. --- Collisionless shocks. % ========================================================= EOF