% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 18.311. Spring semester 2011. Tue and Thu 11:00--12:30, room 2-190. % % ODE or ode: Ordinary Differential Equation(s). Both singular and plural. % PDE or pde: Partial Differential Equation(s). Both singular and plural. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ========================================================================== % ---------------------------------------------------- Lecture 01 Tue-Feb-01 General mechanics of class. Discuss syllabus, grading, books, notes, etc. Outline of the mathematical topics to be covered. Physical phenomena that motivate the material: --- Hydraulic Jumps [kitchen; river floods; flash floods; dams; etc]. --- Shock waves [sonic boom, explosions, super-novas and crab nebula]. --- Traffic flow waves. Others (not mentioned in this lecture) --- Solitary waves [say, in lakes]. --- Diffusion. % ------------------------------------------------------------------------- % --------- NOTE: for the material we start covering now, see % --------- the conservation law notes on the WEB page. % ------------------------------------------------------------------------- % -------------------------------------------------------------------- #001 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: --- What is rho and under what assumptions one can define it. --- What is q and under what assumptions one can define it. --- Point out analogy with continuum hypothesis in fluids and solid matter. The difference between 10^20 and 10^1.5. --- Conservation of cars: Integral and differential forms. % % -------------------------------------------------------------------- #002 Closure problem: more unknows than equations. --- Quasi-equillibrium approximation leading to q = Q(rho). Assumption: there is a stable equillibrium to which system adjusts on a time scale faster than the time scales of interest in problem. --- Discuss generic form of Q. Existence of a maximum throughput q_m = Q(rho_m). % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % ---------------------------------------------------- Lecture 02 Thu-Feb-03 % -------------------------------------------------------------------- #003 Lighthill-Whithams model for Traffic Flow. Recall derivation. The flux function q = Q(rho): --- Discuss practical significance of the existence of q_m = Q(rho_m). Leads to entrance lights at highway access at peak time. Explain purpose, and limitations (no dynamics and not enough space to have cars wait till optimal rho_m is reached). % % -------------------------------------------------------------------- #004 GENERAL FRAMEWORK. 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. (see #011) % SKIP Forestry, molds, bacteria, ... --- examples later. % % -------------------------------------------------------------------- #005 GENERAL FRAMEWORK. How conservation principles lead to pde's: --- Integral and differential form of a conservation law. --- Closure issues and "constitutive equations." Quasi-equillibrium approximations and time-scale limitations. The case of traffic flow and the traffic flow curve. Further examples (below) will clarify these ideas. % % -------------------------------------------------------------------- #006 EXAMPLE: Heat flow along an insulated wire. Describe closure, Fick's law, and do intuitive justification using stat. mech. interpretation of heat and temperature. % % -------------------------------------------------------------------- #007 More than 1-D. Conservation laws in 2-D or 3-D. Flux q is now a vector. Use Gauss theorem to obtain general form rho_t + div(q) = S (S equal sources). % % -------------------------------------------------------------------- #008 EXAMPLE: Heat flow in 2-D or 3-D. Then rho = c_v T, where c_v = specific heat of material, T = temperature, Fick's law applies q = - kappa gradient(T), and kappa = heat conductivity. Thus: Heat equation: T_t = nu Laplacian T, where nu = kappa/c_v = coeff. thermal diffusion. % % -------------------------------------------------------------------- #009 EXAMPLE: Diffusion equation (Salt in water, sugar in coffee, ink in water, ...) Same as heat equation C_t = nu*Laplacian C where C = concentration (salt, ink, sugar, ...) nu = difussion coefficient. % -------------------------------------------------------------------- #010 DIMENSIONS and DIMENSIONAL ANALYSIS. What are the dimensions of kappa? nu? c_v? How long does it take sugar to sweeten a coffee cup without stirring? Idealized problem: start with a very small blob of ink, and ask: What is the radius of the blob of ink, R = R(t), as the blob expands due to diffusion? Dimensional analysis says R(t) \propto sqrt{nu*t} In particular, let L be the size of the coffee cup. Sweatening happens when R = O(L) ===> time = O(L^2/nu). Also the relevant time needed to cool/heat a size L vessel. These times are very long when measured in human-relevant scales. Hence stirring needed. Boiling and convection speed up heating. Questions: why does stirring help? why does convection occur? what would happen when heating something with a flane in the absence of gravity? % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % ---------------------------------------------------- Lecture 03 Tue-Feb-08 % -------------------------------------------------------------------- #011 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. --- Compare q = Q(rho) and q = Q(A) for traffic flow and river flow. Concave versus convex. In one case c = dq/drho decreases and in the other increases. % % -------------------------------------------------------------------- #012 EXAMPLE: Euler Equations of Gas Dynamics in 1-D. Closure, quasi-equillibrium and thermodynamics. Polytropic gas. Body forces (sources). % SKIP 2-D and 3-D. Compressible and incompressible cases. % % -------------------------------------------------------------------- #013 Conservation laws and pde. Higher order (TRANSPORT) effects beyond quasi-equillibrium. Note difference between heat/diffusion and the other examples. In one case the fluxes are functions of the densities only, in the other derivatives of the densities appear. --- Important when gradients are not so small. --- Connection with randomness at microscopic level. % % -------------------------------------------------------------------- #014 % SKIP. Come back to this later. EXAMPLE: Traffic flow and Burger-s like equations. When the gradients are not small, must consider the fact that most people drive ``preventively'' --- i.e.: they 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. % % -------------------------------------------------------------------- #015 EXAMPLE: a first peak at the Navier-Stokes equations. Heat conductivity and viscosity. % % -------------------------------------------------------------------- #016 % SKIP. Students should read it in the WEB page notes. EXAMPLE: Slow granular flow in a silo. % % -------------------------------------------------------------------- #017 % SKIP. EXAMPLE: small transverse vibrations of a string under constant tension T, with motion restricted to a plane. 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; where c^2 = T/rho. --- Note c is a velocity (next lecture will show of what). NOTE: students should check elementary physics book 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. % % -------------------------------------------------------------------- #018 % SKIP. EXAMPLE: small longitudinal vibrations of an elastic rod. 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) = 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 and consider Hooke's law case when T = \kappa*(u_x-1). Alternative derivation using F = m*a on rod differential elements. % % -------------------------------------------------------------------- #019 Recap. Conservation laws and pde. Sources and sinks. Examples: Cars flowing in/out of highway through commuter township. Water flowing into river from small affluents. % % -------------------------------------------------------------------- #020 % SKIP. Conservation laws in more than 1-D. Meaning of the flow vector. Why is it a vector? For some conserved quantity, define the flow q as follows: q is 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. With this definition q is a (scalar) function that depends on the position and the orientation (normal) of the differentials of area (in 3-D) or length (in 2-D) for the surfaces (curves) across which the flux ocurrs. Show that q 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. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % ---------------------------------------------------- Lecture 04 Thu-Feb-10 Do #020 from prior lecture? % ------------------------------------------------------------------------- % --------- NOTE: for the material we start covering now, see the % --------- books by Wan, Haberman, Whitham. % ------------------------------------------------------------------------- Solution of 1-st order (scalar) quasilinear equations by characteristics. Examples: traffic flow and river waves. This will take about 4 lectures. % -------------------------------------------------------------------- #021 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. River waves move faster than the flow. Interpret equation as a statement about a directional derivative of the solution in space-time being zero. 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. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % ---------------------------------------------------- Lecture 05 Tue-Feb-15 % -------------------------------------------------------------------- #022 Examples: Linear problems with constant or simple variable coefficients, where all the calculations can be done exactly. In each example: --- Write characteristics in parametric form. --- Solve and draw the characteristics. --- Eliminate the characteristic variables and find the solution. --- Show where the solution is defined. Example 1: u_t+c_0*u_x = a*u. IVP problem on -inf < x < inf, t > 0. Example 2: x*u_x + y*u_y = y, u(x, 1) = g(x) for -inf < x < inf, y > 0. Example 3: u_x + x^2*u_y = y, with u(x, 0) = g(x) for x > 0. Show that this defines the solution to the right of y = x^3/3. % % -------------------------------------------------------------------- #023 Describe general method, for a problem of the form a(x, y)*u_x + b(x, y)*u_y = c(x, y), u 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. Leads to solution expressed in the form u = u(s, z), with x = x(s, z) and y = y(s, z). where: z = parameter/label for the characteristic curves. s = parameter that results from solving o.d.e.'s above = parameter along each characteristic curve. (s, z) is a curvilinear, coordinate system. Characteristic coordinates. To get the solution must solve for s and z as functions of x and y. Illustrate the coordinates (s, z) graphically. NOTE: in all the examples above, the characteristic curves are independent of the solution. This follows from the equations being LINEAR. That is a, b, and c do NOT depend on u. % % ------------------------------------------------------------------------- % ------------------------------------------------------------------------- % --------- Now we move up to nonlinear problems. %%%%%%%%%%%%%%%%%%%%%%%%% % ------------------------------------------------------------------------- % -------------------------------------------------------------------- #024 Example 4: General kinematic wave u_t + c(u)*u_x = 0, with u(x, 0) = f(x). --- Characteristic form and characteristic speed. --- Solution generally 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 do not exist for all time. --- Show that the characteristics can/do 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. % % ------------------------------------------------------------------------- % % -------------------------------------------------------------------- #025 Review of chain rule in several variables. % % -------------------------------------------------------------------- #026 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. % % -------------------------------------------------------------------- #027 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. 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. Graphical interpretation of shock jump condition: slope of secant line in rho-q=q(rho) diagram. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % ---------------------------------------------------- Lecture 06 Thu-Feb-17 Finish material in #027 Recap of red-light turns green problem. Do green-light turns red problem, both ahead and behind the light. Physical meaning of the various waves observed: Red-turns green --- Path of first in line behind the red light. A characteristic, and also a car path, since u(0) = c(0). --- Location where cars start moving behind the light. Light wave: locus where brake lights turn off. Green-light turns red --- Path of last car through the light. Shock speed and car speed the same when state behind is rho = 0. --- Location where cars stop behind the light. Wave moving backwards from the light. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % ---------------------------------------------------- Lecture 07 Thu-Feb-24 Topics: RECAP of key points (see #027). Consider Traffic Flow with smooth initial wave profile rho = f(x). Or Flood waves """""""""""""""""""""""""""""""""""""""""""" % -------------------------------------------------------------------- #028 Characteristics almost always cross. WRITE THE PRECISE CONDITIONS NEEDED FOR THIS TO HAPPEN. Simple problem: rho_t + q_x = rho_t + c(rho)*rho_x = 0, rho(x, 0) = R(x), c(rho) = dq/drho. Solution by characteristics: x = X(s, t) = C(s)*t + s, (#1) rho = R(s), where C(x) = c(R(x)) = wave speed along initial data. Characteristics do not cross if and only if can solve for s as a function of x and t --- s = S(x, t) --- from (#1) if and only if map s --> x is monotone: X_s not 0. That is: inspect X_s = C'(s)*t + 1. So, if C'(s) < 0 somewhere, there will be a time when X_s = 0. Thus, the condition for crossing is: dc/dx < 0 somewhere in the initial data. Graphics: show how x = X(s, t) = C(s)*t+s looks like as a function of s, for t fixed, as t grows, if C(s) is a localized hump. t = 0 straight line x = s. t >0 moderate straight line develops a wiggle. t >0 large wiggle large enough to produce a local max. and a local min. Hence a range where map is not 1-to-1. Formula for critical time t_c and location x_c where the characteristics cross first, assumming C'(s) < 0 somewhere (no crossings otherwise) Let s_c be the value of s at which C'(s) reaches its largest negative value (i.e.: absolute minimum). Then t_c = - 1/C'(s_c) ... so X_s vanishes. x_c = C(s_c)*t_c + s_c % % -------------------------------------------------------------------- #029 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. % % -------------------------------------------------------------------- #030 First look at EXPANSION FANS: A discontinuity in the initial, or boundary, conditions gives rise to an expansion fan if the edge characteristics do not cross and leave a gap. That is: c is increasing across the discontinuity: EXAMPLES: Red-light turns green. Red-green-red-green ... Will see more later. DEFINITION: An expansion fan is the solution produced by a collection of characteristics, all starting at one single point, but with a range of values for the solution there. % % -------------------------------------------------------------------- #031 First look at SHOCKS: A discontinuity in the initial, or boundary, conditions gives rise to shock if the edge characteristics cross. That is: c is increasing across the discontinuity: EXAMPLE: green-light turns red problem. One shock on each side of light. Shock on the right: last car through light. Shock on the left: locus where cars break behind the light. Shock conditions (see #027). Rankine-Hugoniot shock speed S = [q]/[rho]. Derive from conservation: In shock frame: flow from the left = rho_*(u_ - S) = F_ In shock frame: flow from the right = rho+*(u+ - S) = F+ Then conservation is F_ = F+, which gives S = [q]/[rho]. Graphical interpretation of Rankine-Hugoniot: slope of secant line in rho-q diagram. Note: limit for infinitesimal strength shocks is characteristic speed. % % -------------------------------------------------------------------- #032 Important: crucial to know what is conserved. Note that the equation u_t + u*u_x = 0 can arise from various conservation laws: (a) Conserved density rho = u, flux q = (1/2)*u^2, (b) Conserved density rho = u^2, flux q = (2/3)*u^3 (c) Conserved density rho = u^3, flux q = (3/4)*u^4 .... Each of this gives rise to the same solution as long as there are no shocks, but give rise to DIFFERENT shock speeds, hence different solutions once shocks arise. % % -------------------------------------------------------------------- #033 Important, ENTROPY condition: We need shocks to stop the crossing of the characteristics: characteristics end at shock, and do not continue \ #A on the other side. Hence crossings are avoided. / This requires the condition c_ > S > c+ ``Entropy condition''. Why name? Note that #A implies that INFORMATION IS LOST AT SHOCKS, and problem becomes IRREVERSIBLE once SHOCKS FORM. Hence #A is determining the arrow of time. In physics, information contents is measured by the entropy, that must be non-decreasing (second law of thermodynamics). Hence the name. Example: look at red light turns green problem. In addition to the solution with the rarefaction fan, in principle the problem admits a solution with a ``shock'', but this shock does not satisfy entropy, and in fact generates information (characteristics). This solution is NOT ``physical'' and should be considered ``in-admissible'' in the augmented model. % % -------------------------------------------------------------------- #034 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. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ========================================================================== % ---------------------------------------------------- Lecture 08 Tue-Mar-01 Do third example in #034 % % -------------------------------------------------------------------- #035 Back to entropy condition and R-H jump condition: c_ > S > c+ and S = [q]/[rho]. Inspect graphical meaning of this in rho-q diagram. Consistent for traffic flow and river flows (q concave and convex). Leads to: For traffic flow rho increases across shock. For river flow A decreases across shock. % % -------------------------------------------------------------------- #036 Define linear, semilinear, and quasi-linear 1st order scalar pde. SOLUTION PROCESS FOR KINEMATIC CONSERVATION LAWS THAT SUPPORT SHOCKS (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 used ONLY to prevent characteristics from crossing. 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. Shocks MUST satisfy both the Rankine-Hugoniot jump conditions and the entropy condition. (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. % ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ % IMPORTANT NOTE ABOUT THE "ENTROPY" CONDITION: Applies if one 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): 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. 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 the 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). % ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ % % % -------------------------------------------------------------------- #037 Example: Define Riemann problem, and do it for u_t + ((1/2)*u^2)_x = 0, with u the conserved density, and q = (1/2)*u^2 the flux. Why is Riemann problem important. Talk a bit about Godunov. % % -------------------------------------------------------------------- #038 Example: u_t + ((1/2)*u^2)_x = 0, with u the conserved density, and q = (1/2)*u^2 the flux. Initial Value problem: u(x, 0) = 1 for x > 0 and x < -1 u(x, 0) = -x for -1 < x < 0. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ========================================================================== % ---------------------------------------------------- Lecture 09 Thu-Mar-03 % % -------------------------------------------------------------------- #039 Will do a few more examples with Traffic Flow, in which we consider the special case of a quadratic flow. Hence q = (4*q_m/rho_j^2)*rho*(rho-rho_j). -- Non-dimensionalize the equations and show that they can be reduced to rho_t + q_x = 0, with q = 4*rho*(1-rho). Use interval between light changes to non-dim time. -- Note then c = 4*(1-2*rho) is linear in c. Hence, since rho is conserved, in this case c is also conserved. -- Note that the shock speed in this case is the average of the characteristic speeds across the shock. % % -------------------------------------------------------------------- #040 Example, for #039. a) Red light (at x = 0) turns green at t = 0. Thus rho(x, 0) = 1 for x < 0, ..... c(x, 0) = - 4, and rho(x, 0) = 0 for x > 0, ..... c(x, 0) = 4. b) Green light turns red at t = 1. c) Red light turns green at t = 2. Write full solution, for all times. Track down the shocks that form at x = 0 and t = 1 [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. Do in detail the setting of 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). % % -------------------------------------------------------------------- #041 Example, for #039 a) Red light turns green, with a 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) Let the light turn back to red at t = T > 1. Continue the solution. Identify shocks that correspond to car paths (last car through light) and check shock speed = u for them. c) Let the light turn back to green at t = 2*T. Continue solution. % % -------------------------------------------------------------------- #042 Example, 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; % % -------------------------------------------------------------------- #043 % Left for students to fill in details Example: 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). Notice that, if one knows ``int f(rho) dx'' for all possible f's, one can recover rho (provided rho is, say, continuous). 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. % % -------------------------------------------------------------------- #044 Recap on the laws governing shocks/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. Physics needed to get shocks: flow against gradient and locality. --- Rankine Hugoniot jump conditions for shocks and integral form of the conservation laws. s = [q]/[rho]. Recall simple derivation using conservation in shock frame, where flux is (u-s)*rho. Hence ((u-s)*rho)- = ((u-s)*rho)+. --- Shocks are curves in space time along which characteristics end, so crossing does no occur. They are NOT NEEDED OTHERWISE! Shocks MUST satisfy the ENTROPY condition: c- > s > c+. At a discontinuity where c+ < c-, insert 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. Entropy involves death (of characteristics, at shocks). --- The entropy condition allows causality, with the shock (and the solution on each side) determined by the data (see next). --- Assume piecewise smooth solution with shocks. At a shock, the problem 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. --- The shock conditions in the Q-rho Flow-Density plane. a) Need consistency between entropy condition and jump condition. Jump condition must yield shock velocity satisfying c- > s > c+. 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 cars. Car enter shock region from behind. (Q concave). c) River flow: forward facing shocks, moving faster than water flow. Water enters hydraulic jumps from ahead. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ========================================================================== % ---------------------------------------------------- Lecture 10 Tue-Mar-08 Topics: Shock structure due to "diffusion" type effects. % % -------------------------------------------------------------------- #045 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 % % -------------------------------------------------------------------- #046 The role diffusion plays in stopping steepening and wave breaking. Note that new 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. % % -------------------------------------------------------------------- #047 --- Shock as thin layer where diffusion and nonlinearity balance. --- Shock structure argument: the shock zone in space time is very thin, thus shock look ``locally'' like a plane wave. That is: a traveling wave. % % -------------------------------------------------------------------- #048 Traveling wave for augmented equation. Substitute rho = R((x-s*t)/nu) into the equation. Analyze conditions under which a wave connecting two given states at (x-s*t)/nu exists [this gives shock as nu vanishes] Recover jump and entropy conditions. % % -------------------------------------------------------------------- #049 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. % % -------------------------------------------------------------------- #050 Compute time derivative of \int rho^2 dx for viscous problem. Show decreasing (dissipation). Show time derivative does not go to zero (as nu vanishes) if there are shocks. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ========================================================================== % ---------------------------------------------------- Lecture 11 Thu-Mar-10 Do #049-050 prior lecture BEGIN NEW TOPIC % ======================================================================= % % %%%%%%%%%%%%%%%% Gas Dynamics, Acoustics, and Strings. %%%%%%%%%%%%%%%% % % ======================================================================= % % % -------------------------------------------------------------------- #051 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. --- 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). Derive also Shallow Water equations on a flat bottom. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ========================================================================== % ---------------------------------------------------- Lecture 12 Tue-Mar-15 Finish deriving Shallow Water Equations. (end of #051) Point out same as Gas Dynamics with gamma = 2. % % -------------------------------------------------------------------- #052 Review topic from thermodynamics for ideal gases: Show p = rho^gamma for ideal gas with constant specific heat at constant entropy. % % -------------------------------------------------------------------- #053 Write equations (*) in the form Y_t + A*Y_x = 0. (*) Gas dynamics, constant entropy, so that p = p(rho). Property of p(rho): a^2 = dp/d rho > 0 Show a has dimensions of speed. Compute what a is for Shallow Water. Calculate a for tidal wave in the deep ocean (4.000 m). % % -------------------------------------------------------------------- #054 Gas Dynamics: 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. This happens 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). NEXT ITEM. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ========================================================================== % ---------------------------------------------------- Lecture 13 Thu-Mar-17 Finish with #054. % -------------------------------------------------------------------- #055 % ====================================================================== % % 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 closed (u = 0) and open (R = 0) pipe ends. -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. Better yet, introduce velocity potential: u = phi_x and (a_0^2/rho_0)*R = - phi_t \--------------/ Note this is the pressure perturbation. Hence, second equation is satisfied. Then first equation gives phi_tt - a_0^2*phi_xx = 0. Wave equation. Note boundary conditions: Closed pipe: phi_x = 0. Open pipe: phi = 0. -4- Use (2-3) to write general solution for the wave equation. % % -------------------------------------------------------------------- #056 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) [see #066], but more complicated: 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. % % -------------------------------------------------------------------- #057 Example: simple waves in Gas Dynamics. Assume u-h = constant for initial data. Then characteristics yield u-h = L = constant, as long as characteristic form applies. Hence u = h + L = U(rho) is a function of rho only. The equations then reduces to d/dt (u+h) = 0 along dx/dt = u+a. That is (U+h)_t + (u+a)*(U+h)_x = 0, which is a first order equation of the same type as Traffic Flow and River Flows. Hence characteristics can cross and once this happens we need to re-examine the physics to see what to do beyond breakdown. In this case, again, shocks are the appropriate answer. % % -------------------------------------------------------------------- #058 % SKIP Shocks in Gas Dynamics: Rankine-Hugoniot and Entropy conditions. % % -------------------------------------------------------------------- #059 % SKIP Piston problem. 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. % % -------------------------------------------------------------------- #060 % SKIP (optional material). The p-system --- 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. % % -------------------------------------------------------------------- #061 % SKIP (optional material). Weak shocks in Kinematic Waves. --- Weak shock velocity (average of characteristic speeds). --- Weak shocks and quadratic approximation for flux. % % -------------------------------------------------------------------- #062 % SKIP (optional material). General problem of shock formation in Kinematic Waves. -- Shock formation time and envelope of characteristics. -- Envelopes. Examples of envelopes and shocks. % % -------------------------------------------------------------------- #063 % SKIP (optional material). 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. % % -------------------------------------------------------------------- #064 % SKIP (optional material). Sonic booms (toy model). 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 the plane, so think of the plane as a point source of momentum. -- Knock the problem to 1-D. But then loose the fact that in 3-D the waves, as they move away from the plane, decay [geometrical effect due to 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, the 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. Now -- 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). Still to write. % % ------------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ========================================================================== % ---------------------------------------------------- Lecture 14 Tue-Mar-29 % ---------------------------------------------------- Lecture 15 Thu-Mar-31 Recall result in #056, and do #057. % % -------------------------------------------------------------------- #065 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 = delta p = (a_0^2/rho_0)*R and c = a_0. 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. % EXTRA DETAILS (review of prior material) Linearized gas dynamics: rho_t + rho_0*u_x = 0 and u_t + (c_0^2/rho_0)*rho_x = 0 Show equivalent to wave equation (eliminate either rho or u). Write in characteristic form and solve: 1) u_t + c_0*u_x + (c_0/rho_0)*(rho_t + c_0*rho_x) = 0 ===> rho_0*u + c_0*rho = constant on characteristic x = c_0*t + x_0 2) u_t - c_0*u_x - (c_0/rho_0)*(rho_t - c_0*rho_x) = 0 ===> rho_0*u - c_0*rho = constant on characteristic x = -c_0*t + x_0 Solution superposition of two waves. The wave equation u_tt - u_xx = 0 (set speed c = 1). Convert to characteristic form by introducing v = u_t - u_x. Solve and find general solution to equation. Solution is two waves, moving right and left at speed c = 1. % ================================================================ % % Left in the ink-pot: % % % % Transform wave equation u_{tt} - c^2 u_{xx} = 0 into a 1st order % % system by introducing a potential --- % % interpret equation as a 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. % % % % Show: boundary conditions for a tied string of length l lead to % % a solution of space period L = 2*l --- extend solution by % % "reflecting" across ends]. % % ================================================================ % % % -------------------------------------------------------------------- #066 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. % % -------------------------------------------------------------------- #067 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. % % -------------------------------------------------------------------- #068 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. % % -------------------------------------------------------------------------- % ======================================================================== % % ----------------- Separation of variables/normal modes ----------------- % % ======================================================================== % % -------------------------------------------------------------------- #069 Separation of variables, review. 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: Typical separation of variables solution has the form: u = cos(n*pi*t/L)*sin(n*pi*x/L) which yields u = (1/2)*cos((n*pi*/L)*(x+t)) + (1/2)*cos((n*pi*/L)*(x-t)) using trigonometric equalities. % % -------------------------------------------------------------------- #069 Normal modes. Equations of the form u_t = Lu Relationship with separation of variables: equation invariant under time shift allows separation u = exp(lambda*t) U(x) 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. Explain this works, for example, as long as the assocated eigenvalue problem is self-adjoint: 1) Interpretation of matrices as linear operators. 2) Interpretation of self-adjoint for matrices in terms of the scalar product. 3) Definition of scalar product and Hilbert space. Will not go into details of this: Sturm-Liouville problems. Series and Transforms. When do normal modes work. Etc. % % -------------------------------------------------------------------- #070 % SKIP. Maybe assign in a problem set. 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. % % -------------------------------------------------------------------- #071 % SKIP. Maybe assign in a problem set. Wave equation. Show that: 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 FILE