18.311-MIT. Principles of Applied Mathematics. Spring 2014. MWF 11:00--12:00. Room E17-133. Rodolfo R. Rosales. Lecture Summaries. Here #nnn are references to the Lecture Points file. [PSQ] means Problem Set Question. % ========================================================================== % ---------------------------------------------------- Lecture 01 Wed-Feb-05 General mechanics of class. Discuss syllabus, grading, books, notes, etc. Then do #000. % ========================================================================== % ---------------------------------------------------- Lecture 02 Fri-Feb-07 Do #004, then #001, #002, and #003. % ========================================================================== % ---------------------------------------------------- Lecture 03 Mon-Feb-10 Do #005 and #006. Discuss SOURCES (examples) feeder roads to main artery in traffic flow. (#001) feeder streams in river flows. (#005) body forces in gas dynamics. (#006) Discuss SOURCES (general theory). Do #007 and #010. Note: for now SKIP #008 and #009. % ========================================================================== % ---------------------------------------------------- Lecture 04 Wed-Feb-12 Do #011. #012, #013 + Recap. #010 & intuitive justification of Fick's law. #014. % ========================================================================== % ---------------------------------------------------- Lecture 05 Fri-Feb-14 Finish #014. Time it takes to reach R = 5 cm [salt in water]. Do #015, #016, and jump to #019. *** 1-st order (scalar) quasi-linear equations *** .............. NEW TOPIC The method of characteristics. Examples: traffic flow and river waves. This will take about 6 lectures. % ======================================================================= % % Review of chain rule in several variables. % % It is important that the students be familiar with differentiation % % with several variables, chain rule, and implicit differentiation. % % Please review your 18.02 notes. % % ======================================================================= % % ========================================================================== % ---------------------------------------------------- Lecture 06 Tue-Feb-18 --- Recap solution by characteristics of u_t+c_0*u_x = 0 [linearized traffic flow] and u_t+c_0*u_x = a*u. --- Do examples 2 and 3 in #020 % ========================================================================== % ---------------------------------------------------- Lecture 07 Wed-Feb-19 Do #021, #022. Start with #023. Starting with #023, we move up to nonlinear problems. % ========================================================================== % ---------------------------------------------------- Lecture 08 Fri-Feb-21 Finish #023. Do #024 and #025a. % ========================================================================== % ---------------------------------------------------- Lecture 09 Mon-Feb-24 Recap/finish #025a. Do #025b. Start with #025c. % ========================================================================== % ---------------------------------------------------- Lecture 10 Wed-Feb-26 Recap #025b and finish with #025c. Do #025d. Point out: Shocks have discontinuity "backwards" in traffic flow. Back to "NOTE" in #023 --- consistent with steepening! What do you expect for river flows? Does it match observations? % ========================================================================== % ---------------------------------------------------- Lecture 11 Fri-Feb-28 11a Do #025e [recap of points needed ...]. 11b Then, back to shocks: Shocks in the "green light turns red" traffic flow examples we did have the discontinuity "backwards". This is consistent with steepening. Shocks only needed in this case. Forward discontinuities self-destruct [red light turns green example]. Back to "NOTE" #023a: What do you expect for river flows? Does it match observations? 11c Start with #026. % ========================================================================== % ---------------------------------------------------- Lecture 12 Mon-Mar-03 Recap from #026 "new theory" and do Example 2 there. % ========================================================================== % ---------------------------------------------------- Lecture 13 Wed-Mar-05 a) Note about solving ode's! How do you solve dx/ds = x-y and dy/ds = x+y, x(0) = z and y(0) = 0 .... ? CANNOT DO SEPARATION OF VARIABLES! b) Continue with #026 and do Example 3 there. % ========================================================================== % ---------------------------------------------------- Lecture 14 Fri-Mar-07 a) Finish Example 3 in #026. b) Do 5 in #026 (not allowed discontinuities). % ========================================================================== % ---------------------------------------------------- Lecture 15 Mon-Mar-10 a) Finish with #026 [do Example 1]. b) POINT OUT: method works for linear/semilinear/quasilinear scalar eqns. Define linear/semilinear/quasilinear. c) Start with #027. % ========================================================================== % ---------------------------------------------------- Lecture 16 Wed-Mar-12 Recap main result from #027 and show pictures and computer movies for characteristics and envelopes. 18311 toolkit: demoENV, EIKOdemo, EIKO3Ddemo, and MachCone. For EIKO: brief description of propagation of wave fronts normal to themselves at light speed. Gives rise to rays and caustics. Physical examples for caustics: (1) bright lines at the bottom of a pool in the sun; (2) bright lines on a paper next to a bottle on a table in the sun [clear glass bottle, filled with clear liquid]. (3) In big rooms with stone walls, places where you can hear bits and pieces of conversations from the other side. "Wispering gallery" effect. For MachCone: picture of spherical wave fronts generated by supersonic plane in the past, overlapping at a cone. Cerenkov radiation. % ========================================================================== % ---------------------------------------------------- Lecture 17 Fri-Mar-14 Recap #027. Then compute whole curves in (x, t) plane where (1+t*C'(s)) = 0. Show corresponds to places where rho_x = infinity. Talk a little about envelopes. Recall last lecture: pictures of the envelope of characteristics. Example (Mach Cone): plane at x=y=0 "now", moving along y = 0 at speed c > 1. Then the wave front emitted a time interval t before is at (x+c*t)^2 + y^2 = t^2, for t > 0. [A] [A] plus (x+c*t)*c = t [B] is the envelope equation. --- Use [B] to solve for x in terms of t. --- Plug [B] in [A] and get y in terms of t. --- eliminate t to get x = - sqrt{(c^2-1)/c}*|y|. % ========================================================================== % ---------------------------------------------------- Lecture 18 Mon-Mar-17 Domain of dependence. Review problem Linear 1st order PDE #02. You *cannot* tell where a solution is defined just by looking at the formula. In this problem the answer has a log(y) in it, but this is *not* the reason the solution is not defined for y < 0. The formula does not change if you replace log(y) by log(|y|), and now it makes sense for y < 0. But you cannot use it anyway! Example of an envelope. Family of lines: x = 1.5*s*y - 0.5*s^3. % ========================================================================== % ---------------------------------------------------- Lecture 19 Wed-Mar-19 % ---------------------------------------------------- Lecture 20 Fri-Mar-21 Review/recap of theory so far [#028 through #035]. Highlights only. Expect students to scan these points and *** ask to review points poorly understood. *** Do EXAMPLES from #031 [Examples 3 and 4]. EXAMPLES from #032 [all]. Start with issue of entropy increase at shocks #033, #034, #042. Today: motivate definition of "example entropy" as \int u^2. % ========================================================================== % ---------------------------------------------------- Lecture 21 Mon-Mar-31 Continue with Entropy --- Compute d/dt of \int A^2 [see #042]. --- Do remark in box in #034; and #036 More recap points in #037 -- #043. Students must read AND check that they have no gaps in understanding AND they must do the examples. Ask for review if needed. % ========================================================================== % ---------------------------------------------------- Lecture 22 Wed-Apr-02 Question and answers session. Review envelopes and relationship between conservation laws and shocks. Example 1. Energy conservation equation for shallow water: Not correct to enforce it across hydraulic jumps. C1 Conservation of mass/volume: h_t + (h*u)_x = 0, C2 Conservation of momentum: (h*u)_t + (h*u^2 + p)_x = 0, C3 Conservation of energy E_t + (u*E + p*u)_x = 0, where p = (1/2)*g*h^2 is the average hydrostatic pressure, and E = (1/2)*h*u^2 + (1/2)*g*h^2 is the energy density. C1 and C2 are valid across hydraulic jumps, but C3 is not, since hydraulic jumps dissipate. Note: conservation of momentum makes sense as long as we can neglect the friction forces from the bottom. But, given this, there is no reason to expect extra momentum dissipation at the shocks, which are rather thin. Example 2. 2nd order models of traffic flow. Derive Payne-Whitham, and explain traffic pressure in terms of preventive driving: True velocity drivers aim at is U-mu*rho_x, where mu = mu(rho) > 0. See #044 [note nu = mu*rho] C1 Conservation of cars: rho_t + (rho*u)_x = 0, E2 Proposed acceleration law: u_t + u*u_x = (1/tau)*(U-mu*rho_x-u), where U = U(rho) desired velocity at equilibrium. Equation E2 can be written in the form E3 u_t + u*u_x + (1/rho)*p_x = (1/tau)*(U-u), where p=p(rho) is defined by p^\prime = mu*rho/tau. Here p is called the traffic pressure by analogy with Gas Dynamics. Then E3 leads to the conservation of "momentum" form: C2 (rho*u)_t + (rho*u^2 + p)_x = (rho/tau)*(U-u). However, momentum should not be conserved across traffic shocks, since cars BRAKE there! Similarly, one can also write the following "conservation" laws [none of which is likely to apply for true traffic shocks]. C3 u_t + ((1/2)*u^2 + b(rho))_x = (1/tau)*(U-u), where b^\prime = mu/tau. C4 E_t + (u*E + p*u)_x = (rho*u/tau)*(U-u), where E = (1/2)*rho*u^2 + e(rho), and e = e(rho) is defined by: (e/rho)^\prime = p/rho^2. % ========================================================================== % ---------------------------------------------------- Lecture 23 Fri-Apr-04 Finish with Example 1 in prior lecture, and do Example 2. Do an example from #038/#039/#040, if students ask [did not]. *** Shock structure due to higher order effects *** ............. NEW TOPIC Do #044, #045, #046, and #047. % ========================================================================== % ---------------------------------------------------- Lecture 24 Mon-Apr-07 Do #049. *** Gas Dynamics, Acoustics, and Strings *** .................... NEW TOPIC Start with #050 [recap derivation of equations]. Note that sqrt{dp/drho} and sqrt{g*h} are speeds. Then do: #052 compute wave speeds. % ========================================================================== % ---------------------------------------------------- Lecture 25 Wed-Apr-09 Finish #050 --- [did not do BC in prior lecture]. Do next #053. % ========================================================================== % ---------------------------------------------------- Lecture 26 Fri-Apr-11 Finish with #053. % ========================================================================== % ---------------------------------------------------- Lecture 27 Mon-Apr-14 Start with #054. % ========================================================================== % ---------------------------------------------------- Lecture 28 Wed-Apr-16 Continue with #054 Review of Gauss/Stokes Theorem and implications for solvability of: When is u_x + v_y = 0 equivalent to psi_x = v and psi_y = -u_x? % ========================================================================== % ---------------------------------------------------- Lecture 29 Fri-Apr-18 Continue with #054, specifically: point #054d. % ========================================================================== % ---------------------------------------------------- Lecture 30 Wed-Apr-23 Review of multivariable calculus. Differentiability. Properties of the gradient. Taylor expansion. Implicit function theorem. % ========================================================================== % ---------------------------------------------------- Lecture 31 Fri-Apr-25 Finish with #054d. Characteristics for Klein-Gordon. Characteristic scheme for solving Klein-Gordon, written as u_t + c*u_x = v and v_t - c*v_x = - F(u). Cannot solve exactly, but can use to advance dt, and propagate sln. Use to introduce Domain of dependence and influence. Compact support initial value stays this way. Linear Klein-Gordon by Fourier Transform. Qualitative idea of dispersion (bump disperses because of lost coherence). The example of the rainbow (dispersion). % ========================================================================== % ---------------------------------------------------- Lecture 32 Mon-Apr-28 Do #055, #056, and #057. #055 Brief description of the "numerical" method by characteristics [advance dt only]. Then domains of dependence and influence. #057 Briefly. Explain entropy [what the characteristics do] % ========================================================================== % ---------------------------------------------------- Lecture 33 Wed-Apr-30 *** Separation of variables and normal modes *** ................ NEW TOPIC This is #068, #069, #070, #071, #072. Brief description of separation of variables, and do example u_xx + u_yy = 0 for r < 1 and u given on r = 1. Use polar coordinates, so r(r*u_r)_r + u_theta theta = 0. Point out method works only for some equations in some coordinate systems. Important equations though. Students should read the "short notes on separation of variables" included with problem set #7. Normal modes. Example of heat equation in 0 < x < pi, with zero BC. Start with #069 Linear algebra review: begin with self-adjoint and scalar products. % ========================================================================== % ---------------------------------------------------- Lecture 34 Fri-May-02 Finish linear algebra review [self-adjoint, etc]. Introduce scalar product definition of self adjoint, and use to prove: --- Eigenvalues of self-adjoint matrices are real. --- Eigenvectors of self-adjoint matrices corresponding to distinct eigenvalues are orthogonal. --- NxN self-adjoint matrices have N orthogonal eigenvectors. Use descending technique: reduce size of problem by one by showing the matrix keeps the hyperplane orthogonal to an eigenvector invariant. Then use: --- Any matrix has, at least, one eigenvalue. Follows from "any polynomial has, at least, one root." Point out: scalar product not unique. Examples: --- = Sum_n w_n u_n* v_n, for some weights w_n > 0. One application: cost functions in optimization. --- Any self-adjoint matrix with positive eigenvalues gives rise to a scalar product: u^* A v. Prove this. % % ========================================================================== % ---------------------------------------------------- Lecture 35 Mon-May-05 Generalize to operators last lecture stuff. --- That is: finish #069. Do example in #072. Show examples are self-adjoint. Also, do example WITHOUT any eigenvalues (illustrates the importance of being self-adjoint). Why self-adjoint is important in QM: Energy is real valued and probability is a probability! % % ========================================================================== % ---------------------------------------------------- Lecture 36 Wed-May-07 % ---------------------------------------------------- Lecture 37 Fri-May-09 % ---------------------------------------------------- Lecture 38 Mon-May-12 % ---------------------------------------------------- Lecture 39 Wed-May-14 Do FD schemes. Follow the notes attached to PSet#08, plus examples. Start with lecture in MatLab toolkit GBNS_lecture. Further materiel in the web page [notes section] under --- Stability of Numerical Schemes for PDE's (Quick preview). --- Stability of Numerical Schemes for PDE's. --- Various lecture notes for 18311. Section: Convergence of numerical Schemes. See lecture points #073/#074. Further materiel [in the introduction to problem set #08] Introduction to the vNSA problem series. Example #1 in the notes ...................................... [1] Forwards and backwards differences for u_t+u_x ............... [2] CFL condition. Necessary but not sufficient for stability .... [3] [1] Explain idea behind "artificial viscosity" used to stabilize a scheme. Motivate it by looking at what adding a term u_xx to an equation like u_t+u_x=0 does to the normal modes. For that matter, what it does do u_t+u_xxx=0, or whatever. [2] These are not in the notes. Assigned in pset #08. [3] Both the "good" and "bad" schemes satisfy CFL if dt < dx. % ========================================================================== % %% EOF