LECTURES SCHEDULE FOR 18.311; Spring semester 2010. Room 4-159 ========================================================================== Lecture #01 Tue-Feb-02 General mechanics of class. Discuss syllabus, grading, books, notes, etc. Outline and physical phenomena to be covered: --- Hydraulic Jumps [kitchen; river floods; flash floods; dams; etc]. --- Shock waves [sonic boom, explosions, super-novas & crab nebula]. --- Traffic flow waves. Others (not mentioned in this lecture) --- Solitary waves [say, in lakes]. --- Diffusion. Derive the Lighthill-Whithams model for Traffic Flow rho_t + q_x = 0, where q is a function of rho. Brief discussion of the continuum limit involved, and the assumptions leading to the hypothesis that q = q(rho). ========================================================================== Lecture #02 Thu-Feb-04 NOTE: for the material we start covering now, see the conservation law notes on the WEB page. Recall derivation of the Lighthill-Whithams model for Traffic Flow. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Discrete to continuum limits: --- Densities and fluxes. --- Issues relating to the meaning of the "continuum" limit. Examples: Car densities and flux. Interstellar media (where do super-nova shocks travel on?). Pressure in a gas. Other fluid properties. Other examples: Grade "curve" (idealization versus reality: histograms). River density and flux. Forestry; molds; bacteria. How conservation principles lead to pde's --- recall Traffic Flow. --- Integral and differential form of a conservation law. --- Closure issues & "constitutive equations." Quasi-equillibrium approximations and time-scale limitations. The case of traffic flow and the traffic flow curve. --- Another example: river flow equations. A_t + Q_x = 0. Describe quasi-equillibrium function Q = Q(A). dQ/dA is increasses with A. Justify: at equillibrium forces (friction and gravity) must balance. --- Note that dq/drho (traffic flow) and dQ/dA (river floods) have the dimensions of a velocity. But, the velocity of what? We will answer this question soon. FURTHER EXAMPLES: Euler Equations of Gas Dynamics in 1-D. Closure, quasi-equillibrium and thermodynamics. Polytropic gas. Skip this one. Will come back to it later. Small transverse vibrations of a string under constant tension T, with the motion restricted to a plane (again: the linear wave equation). x = coordinate along string u = u(x, t) transversal displacement of string from equillibrium. rho = density of string (assume constant). Thus rho*u_t = transversal linear momentum density. T*u_x = Transversal component of tension = transversal momentum flux. Use general conservation law machinery now: --- Obtain wave equation: u_{tt} - c^2 u_{xx} = 0; c^2 = T/rho. --- Note c is a velocity (next lecture will show of what). NOTE: the students should check elementary physics book [as used in 8.01/02] where the equation is derived using force [F = m*a] balance on string differential elements. Conservation is a more powerful method, generalizable to many other contexts. SKIP THIS EXAMPLE [Will probably assign in a homework]. Longitudinal vibrations of an elastic rod & the linear wave equation. x = Lagrangian coordinate (particle position at equillibrium). rho = density (mass per unit length) of rod u = u(x, t) displacement. Thus rho*u_t = Longitudinal linear momentum density. T = T(x, t) tension ( T(x0, t) is force by x > x0 on x < x0). Thus -T = momentum flux Elasticity: T is a function of the strain = f(u_x - 1) Explain why u_x-1 is strain. Derive equation & consider Hooke's law case when T = \kappa*(u_x-1). Alternative derivation using F = m*a on rod differential elements. ========================================================================== Lecture #03 Tue-Feb-09 Continue with theme of conservation laws to pde models: Higher order (TRANSPORT) effects beyond quasi-equillibrium. --- Important when gradients are not so small. --- Connection with randomness at microscopic level. EXAMPLES: Traffic flow and Burger-s like equations. When gradients are not so small, the fact that drivers drive "preventively" (try to predict future traffic conditions and proceed accordingly). Simple model: if density is larger ahead (rho_x > 0), slow down below what local density indicates, and conversely. Leads to q = Q(rho) - mu*rho_x, where "mu" is small (what small means will be explained later). Heat flow along an insulated wire. Closure, Fick's law, and intuitive justification using stat. mech. interpretation of heat and temperature. Dimensional analysis: heat equation is T_t = nu*T_xx. Relevant time for cooling/heating of vessel of size L should be L^2/nu. Diffusion: salt in water, for example. Same equation S_t = nu*S_xx where S = salt concentration. A first peak at the Navier-Stokes equations. Heat conductivity and viscosity. Continue with theme of conservation laws to pde models: Adding sources and sinks. Examples: Cars flowing in/out of highway through commuter township. Water flowing into river from small affluents. More than one dimension: --- Show that the flow q corresponding to a conserved density rho, Here q is defined as the amount of rho per unit time and per unit length (area in 3-D) crossing a curve (surface in 3-D) from one side to the other. must have the form q = \hat{n} \cdot \vec{q}, where \vec{q} is a vector, and \hat{n} is the unit normal to the curve (surface) [then q positive means neat flow in the direction of \hat{n}]. Use the standard argument with a rectangular simplex of "infinitesimal" size (a limit is involved), and conservation on an area (volume) that vanishes much faster than the fluxes through the sides --- which, hence, must balance. Examples: --- Heat diffusion in 2 or 3-D. --- Slow granular flow in a silo. Skip this one. Recommend students read the WEB notes. --- Euler equations of fluid dynamics, plus body forces. Compressible and incompressible cases. Skip this one. Will come back to it later. ========================================================================== Lecture #04 Thu-Feb-11 ========================================================================== % The material below is in the books by Wan, Haberman, Whitham, etc. % ========================================================================== Solution of 1-st order (scalar) quasilinear equations by characteristics. Examples: traffic flow and river waves. This will take about 4 lectures. Traffic Flow equations. Traffic density, flow and car velocity. Linearize equation near a constant density and solve u_t+c_0*u_x = 0. Do it first by separation of variables/Fourier. Show solution has form u = f(x - c_0*t). This works because equation is linear and constant coefficients. Need approach that will generalize to u_t + c(u)*u_x = 0. But first, note that: for traffic flow u > c and for river flows u < c. Density waves reach cars from ahead, while river waves move faster than the flow. Interpret equation as a statement about a directional derivative of the solution in space-time being zero. This leads to the solution being constant along the integral curves for the directional derivatives. Definition of characteristics and solution by characteristics. Applies to any equation that can be translated into statements about the directional derivatives of the solutions. Such equations are called hyperbolic. Scalar hyperbolic equations can be reduced to o.d.e.'s along the curves integrating the corresponding directional derivatives. Examples: Start by first looking at examples of linear problems with constant or (simple) variable coefficients, where all the calculations can be done exactly. Example 1: Solution of u_t+c_0*u_x = a*u. Example 2: x*u_x + y*u_y = y, with u(x, 1) = g(x) for -inf < x < inf Write characteristics in parametric form. Solve and draw the characteristics. Then eliminate the characteristic variables and find the solution. Show where the solution is defined. ========================================================================== Lecture #05 Thu-Feb-18 Continue with the solution of 1-st order (scalar) quasilinear equations by characteristics. Example 3: u_x + x^2*u_y = y, with u(x, 0) = g(x) for x > 0. Write characteristics in parametric form. Solve and draw the characteristics. Then eliminate the characteristic variables and find the solution. Show where the solution is defined. Describe again general method, for a problem of the form a(x, y)*u_x + b(x, y)*u_y = c(x, y) where u is given along some curve. That is: u = U(z) on some curve x = X(z) and y = Y(z). Equations: dx/ds = a, dy/ds = b, and du/ds = c, to be solved with the conditions x = X(z), y = Y(z) and u = U(z) for s = 0. In the end solution ends up expressed in the form u = u(s, z), with x = x(s, z) and y = y(s, z). where: z = parameter/label that tells us which characteristic curve we are in. s = parameter that results from solving o.d.e.'s above. (s, z) is a new, curvilinear, coordinate system. To get the solution must solve for s and z as functions of x and y. All the examples above are linear, and thus have the property that the characteristic curves are independent of the solution. Now we move up to considering a nonlinear problem: Example 4: General kinematic wave u_t + c(u)*u_x = 0, with u(x, 0) = f(x). Characteristic form. Characteristic speed. Solution now cannot be written explicitly. Geometrical interpretation of the solution --- leads to a clear picture of how conservation is achieved: "sliding slabs" image. Wave distortion/steepening and wave breaking. Smooth solution dos not exist for all time. Show characteristics can cross in space time. Derivatives become infinity at time of first crossing. Note: dc/du < 0 for traffic flow, and dc/du > 0 for river flows. Consequences for wave steepening. ========================================================================== Lecture #06 Tue-Feb-23 ========================================================================== Lecture #06 Tue-Feb-23 Review of chain rule in several variables. Back to nonlinear kinematic waves and wave-breaking: After wave breaking, need more physics to fix the problem. Explain: why does the model fail there? (quasi-equilibrium). Fix to the problem is NOT in the mathematics. Need NEW physics! Will study traffic flow to learn how to fix it. Back to traffic flow, and example problems there. --- Red light turns green. Gap in characteristic field. Argue nice dependence on perturbations to data. Fill gap by taking limit of smeared discontinuity. Obtain expansion fan solution c = x/t. Explicitly solve for rho in the case where q = q(rho) is quadratic. --- Green light turns red. Split into two initial-boundary value problems. Do case of "light" traffic first. Show that, ahead of the light, the characteristics cross. ========================================================================== Lecture #07 Thu-Feb-25 Continue with the green light turns red example in traffic flow. Light traffic case. Ahead of light: characteristics cross. Resolve issue by introduction of the last car to make the light. Discontinuity in solution where characteristics are chopped. Speed of this discontinuity is obvious. Behind the light: again, characteristics cross. Argue that in real life drivers wait till the last moment to break. Another discontinuity needed [location of thin layer where cars break]. How does one compute it's velocity? Get law speed = [Q]/[\rho] by arguing "flow of cars into discont. = flow of cars out" Note flow = rho*(u-s). Note limit as [rho] vanishes is characteristic speed. Again: characteristics chopped at discontinuity and crossing avoided. ========================================================================== Lecture #08 Tue-Mar-02 RECAP of key points. Consider Traffic Flow with smooth initial wave profile rho = f(x). Or Flood waves """""""""""""""""""""""""""""""""""""""""""" (1) Characteristics cross almost always. WRITE THE PRECISE CONDITIONS NEEDED FOR THIS TO HAPPEN (Namely: dc/dx > 0 somewhere in the initial data) (2) Evolution of wave profile, as given by the characteristic solution. Graphical interpretation: --- Move each point on graph at velocity c(rho). Evolution as sliding of horizontal slabs at different velocities (guarantees conservation). --- Causes wave steepening, wave breakdown, and multiple values. --- Mathematical model breakdown. Quasi-equillibrium assumption fails and PDE model breaks down. --- Back to "physics". Need to augment model with new physics, namely: shocks in the case of traffic flow or flood waves. (3) Discontinuities in the initial conditions give rise to: Expansion fans if edge characteristics do not cross and c is increasing from "left" density value to "right" density. EXAMPLE: red-light turns green. give one with signaling. DEFINITION: solution produced by a collection of characteristics, all starting at one single point, but with a range of values for the solution there. Shocks if edge characteristics cross. EXAMPLE: green-light turns red problem. One shock on each side of light. EXPANSION FAN EXAMPLES. Solve: u_t + u*u_x = 0, with u(x, 0) = 0 for x < 0, and u(x, 0) = 1 for x > 0. u_t + u*u_x = -u, with u(x, 0) = 0 for x < 0, and u(x, 0) = 1 for x > 0. u_t + u*u_x = 0, with u(x, 0) = 1 for x > 0, u(0, t) = 1 for 0 < t < 1, and u(0, t) = 0 for 1 < t. (4) SOLUTION PROCESS FOR KINEMATIC CONSERVATION LAWS (and other scalar linear or semi-linear or quasi-linear p.d.e.) ------------------------------------- \DEFINE THIS/ (A) Solve for the characteristics starting at EVERY point along the curves with data (initial, boundary, whatever). If the data is given with different formulas for various segments, solve each segment separately. (B) Check for GAPS in the characteristics, caused by jumps (discontinuity in the data), and fill-in the corresponding fans of characteristics. (C) Inspect the set of characteristics thus obtained and check for crossings. Use shocks to eliminate the crossings (with the techniques to be described below). ** Shocks are used ONLY to prevent characteristics ** from crossing, and characteristics converge into ** them and STOP there! (D) After inserting the shocks, and solving for their paths in space time, check again characteristics. Make sure that all crossings have been resolved, and that (for each shock) you do have the characteristics on each side ending there. (E) Note that a shock may pass through different "regions" where the solution on each side is given by changing formulas (characteristics starting at different parts of data). Keep this in mind when solving for their paths. (F) Check for possible shock "collisions" and resolve them. ========================================================================== Lecture #09 Thu-Mar-04 LAWS GOVERNING SHOCK/HYDRAULIC JUMPS IMPORTANT: Shocks are not always the answer to wave breakdown and multiple values. Shocks introduce new physics! Without shocks, system is reversible. With shocks, system is not reversible. Explain what is needed to get them: flow against gradient and locality. --- Rankine Hugoniot jump conditions for shocks and integral form of the conservation laws. s = [q]/[rho]. Rederive using conservation in shock frame, where flux is (u-s)*rho. Hence ((u-s)*rho)- = ((u-s)*rho)+. --- Shocks as curves along which characteristics end, so crossing does no occur. NOT NEEDED OTHERWISE! Allowed and NOT-allowed discontinuities. ENTROPY condition: c- > s > c+. When violated need an EXPANSION FAN, not a shock! Arrow of time and causality: characteristics die at shock, hence INFORMATION IS LOST. Irreversible evolution, FORWARD IN TIME ONLY. Show how entropy allows causality, with shock determined by data. Show how a time-reversed shock violates entropy. Entropy involves death (of characteristics, at shocks)! --- Assume piecewise smooth solution with shocks. At a shock problem then splits into three subproblems: - Solution ahead of the shock. Smooth and satisfying characteristic form. Determined at each point by a single characteristic connecting to data. - Solution behind the shock: same deal as solution ahead of shock. - Shock position determined by solving o.d.e. given by jump cond. --- Graphic depiction of shock conditions in Q-rho Flow-Density plane. a) Need consistency between entropy condition and jump condition. Jump condition must yield shock velocity satisfying c- > s > c+. This works for for Q concave or convex. More complicated otherwise. May get back to this later. FOR NOW WE ONLY CONSIDER SITUATIONS WHERE THE FLUX IS CONVEX OR CONCAVE. b) Traffic flow: backward facing shocks, moving slower than the cars. Car enter shock region from behind. (Q concave). c) River flow: forward facing shocks, moving faster than the water flow. Water enters hydraulic jumps from ahead. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ IMPORTANT NOTE ABOUT THE "ENTROPY" CONDITION: They apply for assumes standard drivers and standard driving conditions. Using "special" drivers you can arrange to have discontinuities that do not satisfy the entropy condition: e.g.: --- Example (traffic flow): consider the situation at the start of a car race, with all the racing cars neatly organized in a pack behind a lead car. This gives rise a "square" wave density shape, and the entropy condition is violated by the front discontinuity (at least). But this is "car ballet", not traffic flow. --- Example (traffic flow): driver that goes slower than road conditions allow, creating long line of cars behind [common in mountain roads]. Again, requires special driver to maintain the discontinuity at the front of the pack. The discontinuity at the end of the pack is a standard shock, though. --- Example (river flow): push water from behind with a paddle (this is the equivalent of the second example above in traffic flow). ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ In response to question by student: Can one calculate how much "information" is lost at a shock? Answer depends on how one measures the amount of information that one has. However, a "partial" answer to this question goes as follows 1) As long as the solution has derivatives, rho_t + q(rho)_x = 0 implies the conservation of any f(rho), with flux F(rho) given by the equation F' = c*f'. 2) When shocks appear, this is no longer true. Hence one can measure the "loss" caused by a shock by picking an "appropriate" f(rho). 3) Example: A_t + ((1/2)*A^2)_x = 0, with A conserved [baby example of a river floods equation]. Show then d/dt (\int_a^b (1/2)*A^2 dx) = (1/3)*A^3(a, t) - (1/3)*A^3(b, t) - (1/12) [A]^3 \-------------------------------/ \-------------/ Conservative part (a flux) Stuff lost at shock. where [A] = A_ - A+ > 0 is the jump across the shock in A. Note that \int A^2 is a rough measure of the amount of information, in the following sense: for fixed total area \int A in some interval, and A > 0, the minimum value for \int A^2 occurs when A is constant. The more structure (info) the function A has, the larger \int A^2 will be. ========================================================================== Lecture #10 Tue-Mar-09 Examples of traffic flow problems. Take special (non-dim) case with Q = 4*rho*(1-rho) and solve/show --- Shock speed average of wave speeds on each side. --- Then solve problem: a) Red light (at x = 0) turns green at t=0. Thus rho(x, 0) = 1 for x < 0, and rho(x, 0) = 0 for x > 0. b) Green light turns red at t = T. c) Red light turns green at t = 2*T. Write full solution, for all times. Track down the shocks that form at x=0 and t=T [last car through light and stopping locus on x < 0] as they interact first with the rarefaction fan from x = t = 0, and later with the one from x = 0 and t = 2*T. Go in detail over setting up the ode's for the shocks (from the Rankine Hugoniot conditions) and solving them. Check that the entropy conditions are satisfied, and that the shocks do not interact with any other waves that may change their path (for example: with the first car through the light at t = 0). % %% OTHER EXAMPLES. Not done (maybe they will be in problem sets) %%%%%%%% %--- a) Red light turns green, with finite line of cars stopped behind % the light: rho = 0 for 0 < x; % rho = 1 for -2 < x < 0; % rho = 1/2 for x < -2; % Solve using characteristics. % Then plug in expansion fans and shocks. % b) In prior situation, let light turn red at t = T > 1. % Continue solution. Identify shocks that correspond to car paths % (last car through light) and check shock speed = u for them. % c) Turn light back to green at t = 2*tau. Continue solution. % %--- Signaling problem. % Solve u_t + u*u_x = 0 on x > 0 with u(0, t) = 1 for t < 0; % u(0, t) = 1/2 for t > 0; % %% END OF OTHER EXAMPLES. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Shock structure due to "diffusion" type effects. --- Diffusion effects in traffic flow: "look ahead" by drivers. Simple model for this: q = Q(rho) - nu*rho_x Leads to rho_t + (Q(rho))_x = nu*rho_xx --- The role diffusion plays in stopping steepening and wave breaking. Note that new added term adds a contribution to the car flux which is of the same type as the one diffusion (as in heat) causes. As wave steepens, this term gets larger and larger, till it eventually can stop the steepening. ========================================================================== Lecture #11 Thu-Mar-11 Continue with: Shock structure due to "diffusion" type effects. --- Shock as thin layer where diffusion and nonlinearity balance. --- Shock structure: Make argument that shock zone in space time is very thin, so shock looks like a plane wave. --- Traveling wave for augmented equation. Recover jump and entropy conditions. --- The quadratic flow case. Wave speed c is linear in rho. Show that then rho_t + (Q(rho))_x = nu*rho_xx is the same as c_t + c*c_x = nu*c_xx Burgers's Equation. Get explicit solution for traveling waves. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % -------------------------------------------------------------------------- NEW TOPIC ------ Gas Dynamics, Acoustics, and Strings. ------------------ -------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Gas Dynamics: --- Derive equations in 1-D (using conservation of mass and momentum), plus the quasi-equilibrium isentropic assumption p = p(rho). Example: p = kappa*rho^gamma. ========================================================================== Lecture #12 Tue-Mar-16 Continues with Gas Dynamics --- Boundary conditions at the end of the pipe: Closed pipe u = 0; and Open pipe p = p_0; --- For smooth solutions, manipulate equations into the form rho_t + u*rho_x + rho*u_x = 0 u_t + (a^2/rho)*rho_x + u*u_x = 0 where a^2 = dp/drho Calculate a for ideal gas case. Check dimensions: a is a velocity (sound speed, as we will see). --- Notice form Y_t + A(Y)*Y_x, similar to the scalar case, but with the wave velocity replaced by a matrix. Look for o.d.e. forms [i.e. characteristics] by doing linear combinations of the equations. Need to find combinations that produce only one directional derivative in space-time. HYPERBOLIC: equation can be reduced to statements about directional derivatives of the solution. Show it works if using eigenvalues/eigenvectors L*A = c*L: Characteristic form: L*(Y_t + c*Y_x) = 0, or L*dY/dt = 0 along dx/dt = c. Then, along the curves dx/dt = c, solution behaves (sort of) like an o.d.e. Hyperbolic: have enough linearly independent (real) eigenvectors (in this case, 2) so that equation is equivalent to stuff above. If and only if A is real diagonalizable. Before applying these ideas to the full Gas Dynamics problem, LINEARIZE near the equilibrium solution u = 0, and rho = rho_0, and analyze the resulting problem (this is ACOUSTICS). ========================================================================== % The material below can be (partly) found in the book by Wan. % ========================================================================== ---------------- Linear Gas Dynamics - Acoustic in a pipe ---------------- -1- Write equations: R_t + rho_0*u_x = 0 and u_t + (a_0^2/rho_0)*R_x = 0, where R is the density perturbation to rho_0 [rho = rho_0 + R] Boundary conditions for open (u = 0) and closed pipe ends (R = 0). -2- Find eigenvalues and eigenvectors c = +/- a_0 and L = (+/- a_0, rho_0) Write in characteristic form and solve. Solution superposition of two waves: left and right going sound waves. -3- Show equivalent to wave equation (eliminate either R or u). What do boundary conditions for open/closed yield for wave equation. -4- Use (2-3) to write general solution for the wave equation. ========================================================================== Lecture #13 Thu-Mar-18 Continue with acoustics and the wave equation: Characteristic form for wave equation: u_tt - c^2*u_xx = 0. WAY #1: use relationship with acoustics and its characteristics. Equivalently: Transform equation into 1st order system introducing a potential. --- Equation is curl(V) = 0 for vector field V = [u_t, c^2 u_x]. --- Remind/prove theorem: Irrotational vector fields in simply connected domains are gradients. --- Remind students to review the Green/Gauss/Stokes theorems in calculus. --- HENCE u_t - \phi_x = 0 and \phi_t + c^2 u_x = 0, which is acoustics with phi = R. WAY #2 More symmetric: Look for linear combinations of u_t and u_x that yield equations with directional derivatives only. This gives: (u_t + c*u_x)_t - c*(u_t + c*u_x)_x = 0 (u_t - c*u_x)_t + c*(u_t - c*u_x)_x = 0 Solve this and get again solution. Anther example: u_tt - c^2*u_xx + V(u) = 0, which yields (u_t + c*u_x)_t - c*(u_t + c*u_x)_x = - V(u), (u_t - c*u_x)_t + c*(u_t - c*u_x)_x = - V(u). Two families of characteristics, that interact with each other. Can no longer produce an exact solution (i.e.: cannot solve explicitly for u_t + u_x, or u_t - u_x, along their corresponding characteristics. However: 1. Examine how the solution advances in space time. 2. Notions of domain of dependence and domain of influence. 3. Light/sonic cone. No information can propagate faster than c. If solution vanishes outside an interval, it stays zero outside light cone later. ---------------------------- Waves on strings ---------------------------- Derive wave equation for small transversal motion of an homogeneous string under tension, using conservation of transversal momentum. (rho*u_t)_t - (T*u_x)_x = 0. rho = mass per unit length .... Thus rho*u_t = momentum density. T = tension ................. Show how -T*u_x is the momentum flux (force from xa). Hence: u_tt - c^2*u_xx = 0, where c = sqrt(T/rho). Check units are correct. If string tied at ends: u vanishes there. If string loose at ends: u_x vanishes there. Put string on an elastic bed. This adds a restoring force at each point. Equation becomes u_tt - c^2*u_xx + b*u = 0, where b = k/rho and k is the elastic restoring force constant. ========================================================================== Lecture #14 Tue-Mar-30 ------------------ Separation of variables/normal modes ------------------ Another way to solve the wave equation (works for other equations too). Explain idea of separation of variables, write solution as product, etc. Example: do heat equation in 1D with T=0 at ends. do wave equation in 1D with u_x = 0 at ends. Note: for the wave equation the solutions obtained in this way should be compatible with the form u = f(x-c*t) + g(x+c*t). Exercise. Relationship with normal modes (equation invariant under time shifts). Example: write wave equation as u_t = v and v_t = u_xx. heat equation u_t = u_xx --- Note analogy with linear o.d.e. dY/dt = A*Y, A NxN matrix, solved by finding eigenvalues and eigenvectors of A. --- Hence look for solutions of the form u = e^{\lambda*t} v(x) Solve and find normal modes (eigenvalues and eigenfunctions). General solution: Superposition ... leads to Fourier Series, etc. Example: heat equation with various B.C. 1. In a ring: periodic. 2. Zero T at ends. 3. Zero flux at ends. Various types of Fourier series. Will not go into this: Sturm-Liouville problems. Series and Transforms. When do normal modes work. Etc. % ---------------- ASSIGN IN A PROBLEM SET -------------------------- % @#@-P Consider a string tied at the ends. Use a-dimensional variables. Then: u_tt - u_xx = 0 and u = 0 at x = 0 and x = L Find normal modes (or separate variables), and find connection with characteristics: Normal modes as superpositions of a right and a left traveling wave. % ------------------------------------------------------------------- % % ------------------------------------------------------------------- % % %%%%%%%%%%%%%%%%%%%% BACK TO FULL GAS DYNAMICS %%%%%%%%%%%%%%%%%%%% % Derive characteristics for the full 1-D isentropic Gas Dynamics. rho_t + u*rho_x + rho*u_x = 0 u_t + (a^2/rho)*rho_x + u*u_x = 0 where a^2 = dp/drho. Write in the form Y_t + A(Y)*Y_x = 0, and find the eigenvalues and left eigenvectors for A (namely: solve L*A = c*L). Then c = u +/- a, and the characteristic form is: +/- (a/rho)*(drho/dt) + du/dt = 0 along dx/dt = u +/- a. Two sets of characteristics, which interact and couple. Situation similar to u_tt - u_xx + V(u), but more complicated because now the characteristic speed is no longer constant. Characteristics in the same family may cross ... leading to shocks. Introduce h = h(rho) by property dh/drho = a/rho. Show for ideal gas h = 2*a/(gamma-1). Then d/dt (u +/- h) = 0 along dx/dt = u +/- a. i.e (u +/- h) is constant along characteristics. Show how this, in principle, determines the solution. At each point in space time two characteristics [C+ and C-], each carrying information from a different part of the initial data, which combined gives the solution at the point. But now the characteristics are neither straight, nor can we solve for them explicitly, because they interact with each other. Example: simple waves. If time permits: Rankine-Hugoniot and Entropy condtions for shocks. % %%%% PISTON PROBLEM. SKIP. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Use Riemann invariant form of equations in simple problem: --- Gas in a pipe, initially at rest: rho = rho_0 and u = 0, x > 0. --- Some signal at x = 0, t > 0 (sound) u = sigma(t) --- Argue left Riemann invariant constant through-out flow, provided no shocks. Reduce problem to finding right R. I. --- Point out problem is now very similar to Traffic flow. --- Do case sigma(t) = V < 0 (constant). Show need for expansion fan in C+ characteristics --- Do case sigma(t) = V > 0 (constant). Show C+ characteristics cross. Need shock, and left Riemann invariant assumption has to be re-examined. % %%%% END OF PISTON PROBLEM. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % %%%% OPTIONAL MATERIAL. SKIP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The p-system @#@-P --- Characteristic form. Riemann invariants and SIMPLE waves. Shocks. --- Isentropic Gas Dynamics in Lagrangian Coordinates is the p-system (change of variables). --- p-system same as equation for a nonlinear vibrating elastic rod. Weak shocks in Kinematic Waves. @#@-P --- Weak shock velocity (average of characteristic speeds). --- Weak shocks and quadratic approximation for flux. General problem of shock formation in Kinematic Waves. --- Shock formation time and envelope of characteristics. --- Envelopes. Examples of envelopes and shocks. Burgers' equation. --- Cole-Hopf transformation: exact solution. --- Solution of heat equation by Green's function and application to Burgers. --- The small viscosity limit in Burgers'. Laplace's method. Wave equation. @#@-P Boundary conditions for a tied string of length L lead to a solution of space period P = 2*L --- extend solution "reflecting" across ends. % %%%% END OF OPTIONAL MATERIAL. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % %%%% SONIC BOOMS. SKIP. See notes to appear in WEB page %%%%%%%%%%% % Sonic Booms Full problem requires 3-D and motion of finite body through air. Very messy. Simplify dramatically, to bare bone effects. Use axe if needed: --- Not interested in details near plane, so think of plane as a point source of momentum. --- Knock problem to 1-D. But then loose fact that in 3-D the waves, as they move away from the plane, decay [geometrical effect of expansion]. Replace this by adding some damping into the system. --- Replace Gas Dynamics by the simplest (nonlinear) model with wave velocity dependence on the solution, e.g. use a simple wave model. Thus, TOY MODEL is p_t + ((1/2)*p^2)_x = delta(x-V*t) - a*(p-p_0), a > 0. where p_0 = "ambient" state into which plane moves. --- Use a Galilean transformation to set p = p_0 --- Explain meaning of delta function (point source of p) and derive jump conditions across source path x = V*t. --- Write characteristic equations for problem away from x = V*t. Assume initial conditions p = p_0 = 0, and solve problem: Case V < 0. -1- Solution behind plane trivial. -2- Show characteristics starting behind the plane reach it, go through it, and determine solution ahead of plane. These do not cross, and set-up a steady state ahead of plane (show solution by characteristics yields p = p(x-V*t) ahead of plane. -3- There is a shock separating -2- from the air at rest far ahead of the plane. But this shock has strength that vanishes as it moves further and further away from plane. -4- Solution is smooth steady state in t --> infinity limit. Case V > 0 (but not too large) -5- Show need expansion fan behind the plane. -6- Steady state solution analog to 2 has characteristic crossing and needs a shock. -7- Solve for steady state and show solution has a square root singularity. -8- Steady state, t ---> infinity solution, now has shock riding ahead of plane. Case V > 0 "large" (plane catches up to shock ahead). Not done. % %%%% END OF SONIC BOOMS. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % The material below is (partly) contained in the WEB page notes: % % Stability of Numerical Schemes for PDE's. % % Further stuff can be found in the introductions to the problem suites % % (also to be posted in course WEB page) % % --- GBNS Good and Bad Numerical Schemes. % % --- vNSA von Neumann Stability Analysis. % % --- AENS Long wave equation associated to a numerical scheme. % ========================================================================== ========================================================================== Lecture #15 Thu-April-01. Lecture #16 Tue-April-06. Lecture #17 Thu-April-08. Finite difference schemes for PDE's. Follow notes in WEB page: --- Stability of Numerical Schemes for PDE's. 1 Naive Scheme for the Wave Equation. ............ lecture #1 2 von Neumann stability analysis for PDE's. ...... lecture #2 3 Numerical Viscosity and Stabilized Scheme. ..... lecture #3 ALSO USE the MatLab script in the 18311 toolkit: GBNS_lecture. Reminder of details to include here: 1) von Neumann stability analysis Recall that solutions to time-evolution linear PDE's can be found by separation of the time variable --- analogy with ODE approach --- leading to an eigenvalue problem. Extend this idea to constant coefficients linear finite difference schemes -- key to the von-Neuman stability analysis. 2) Examine instabilities using associated equation: --- Explain behavior via forward and backward heat equations. --- Introduce stabilization by artificial viscosity (general idea). 3) Define CONSISTENCY. von Neumann and consistency: Numerical and exact growth rates; comparison in the small k limit. 4) Define STABILITY. 5) Lax Theorem: for linear schemes, Consistency+Stability ==> Convergence. ========================================================================== ========================================================================== ========================================================================== Topics for further lectures (5 to be skipped). ========================================================================== 1 Lecture More examples of von Neuman stability analysis; associated equation, and stabilization by artificial viscosity: Lax-Friedrich sheme. ========================================================================== 1 Lecture Introduction of the Discrete Fourier Transform (DFT). ========================================================================== 1 Lecture Continuum limits: Fourier Series and Fourier Transforms (and their inverses). ========================================================================== 1 Lecture Relationship with Laplace Transforms. Inverse of the Laplace Transform. ========================================================================== 2 Lectures Fourier and derivatives; convergence rates. Fast Fourier Transform. Spectral and pseudo-spectral p.d.e. methods. ========================================================================== 1 Lecture Mechanical Vibrations. Linear and nonlinear spring lattice equations. Continuoum limits. ========================================================================== 1 Lecture Slinkies and measuring of the lattice parameters. ========================================================================== 1 Lecture Failure of thermodynamics equillibration in 1-D lattices. Fermi-Pasta-Ulam problem. ========================================================================== 1 Lecture Numerical solution of the lattice equations for nonlinear springs. Nonlinearity produces formation of constant deformation regions, joined by corners. Corners = discontinuities in deformation rate = shocks. ========================================================================== 1 Lecture Continuoum limit equations for a nonlinear spring lattice = equations for a nonlinear vibrating rod = nonlinear wave equation = p-system. Characteristics and Riemann invariants. Two wave families. Theoretical explanation of the corner (shock) formation. ========================================================================== 1 Lecture Envelopes, Mach Cones, and Sonic Booms. ========================================================================== 1 Lecture Thermodynamic considerations and higher order effects: viscosity. The role viscosity plays in stopping steepening and wave breaking. Weakly nonlinear, weakly viscous waves in Gas Dynamics and Burgers Equation. ========================================================================== % %% EOF