Spring 2009
http://www.amath.washington.edu/courses/586-spring-2009

MWF 2:30-3:20: Loew 206

This class is being offered on-line through EDGE, which provides live streaming of each class and archived lecture videos within two hours thereafter. See http://www.engr.washington.edu/EDGE/amath586/amath586vd.html.

 

Instructor: Prof. Chris Bretherton breth@washington.edu ATG 704, x5-7414 Office hours: MWF 1:30-2:20, or by appointment. TA: Alan Chen amath586@gmail.com Office hours: MW 11:30-12:30, Guggenheim 416

Course Description

Numerical methods for time-dependent ordinary and partial-differential equations, including explicit and implicit methods for hyperbolic and parabolic equations. Stability, accuracy, and convergence theory. Spectral and pseudospectral methods

Prerequisites

Notes and Recommended Text

Scanned lecture notes will be posted. However, for a more comprehensive treatment, I recommend the following texts:

• Durran, D.R., 1999: Numerical methods for wave equations in geophysical fluid dynamics. Springer-Verlag, 465 pp. (We will loosely follows Chapters 1-5; will be cross-referenced in lecture notes).
• LeVeque, R. J., 2007: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 339 pp. (was used for AMath 585, and has much of the material I cover in 586, but not cross-referenced to lecture notes)

Syllabus

Grading

• Homework (60%), posted to class web page and assigned on a quasi-weekly basis. Homework is heavily oriented toward implementation of numerical methods in Matlab and testing of their theoretical properties. Consultation with your fellow students is encouraged, but everyone needs to write out the homework in their own words, and write their own Matlab scripts. NEW: The TA will respond within 24 hours to homework questions emailed to amath586@gmail.com with a cc to breth@washington.edu. Only one email per student per calendar day, please. If there is no response, please email breth@washington.edu with a cc to amath586@gmail.com and the instructor will try to answer your questions. On-campus students shuld submit their homeworks in class or to the TA's mailbox in Amath. Supporting Matlab scripts may be printed out or submitted to the TA via email at amath586@gmail.com to save paper. EDGE students, please submit the homework to the TA as a word-processed file by email, including your Matlab files as attachments. Late homework (after 11:59pm on due date) will be accepted for up to three subsequent days, but there will be a late penalty of 25% on all but the first late assigment.
• Midterm assignment(20%) and final assignment(20%). Like regular homework, but no consultation with anyone and no late assigments accepted.

Schedule

No class:

• We-Fr 20-22 May
• Mo 25 May: Memorial Day (vacation)

Homework and Exams

Item	Due Date	Download Solutions
Homework #1	due Mo 13 Apr	HW #1 solutions
Homework #2	due Mo 20 Apr	HW #2 solutions
Homework #3	due Mo 27 Apr	HW #3 solutions
Homework #4	due Mo 4 May	HW #4 solutions
Take-home midterm	due Mo 11 May	Midterm solutions
Homework #5	due Mo 18 May	HW #5 solutions
Homework #6	due Fr 29 May	HW #6 solutions
Take-home final	due 5 pm Fr 12 Jun	Final solutions

Prof. Bretherton is on travel until We 24 June attending two week-long back-to-back conferences. Graded finals handed in on paper will be available outside Prof. Bretherton's office ATG 704 starting then, and final course grades will be transmitted to the registrar then as well. Sorry for the unavoidable delay.

Lecture Notes and Handouts

• Mo Mar 30: Archetypical time-dependent PDEs, well-posedness, finite difference vs. series expansion methods.
• We Apr 01: Finite difference operators, local truncation error/order of accuracy, implementation of BCs.
• Fr Apr 03: Space-time differencing, accuracy+stability= convergence, energy-integral stability analysis of upwind FDA to advection equation.
• Mo Apr 06: Von Neumann stability analysis
• We Apr 08: Sufficiency of VN analysis; CFL condition
• Fr Apr 10: Aliasing, amplitude/phase error, discrete dispersion relation.
• Mo Apr 13: Time differencing and amplification equation.
• We Apr 15: More single-stage time-differencing methods and their amplification characteristics.
• Fr Apr 17: Stability of forward/backward/trapezoidal time differencing for diffusion equation. 4th order Runge-Kutta.
• Mo Apr 20: 2,3 order Adams-Bashforth methods and their stability. Intro to leapfrog.
• We Apr 22: Leapfrog stability and damping the computational mode; start space differencing generalities
• Fr Apr 24: More space differencing generalities for advection equation; inevitability of numerical diffusion/dispersion.
• Mo Apr 27: Lax-Wendroff method; conservation laws.
• We Apr 29: REA finite volume methods.
• Fr May 01: Monotonic slope-limiter REA methods.
• Fr May 01: Handling multiple space dimensions, source terms, and nonconstant coefficients. Splitting.
• We May 06: Fourier spectral methods I. Basic theory
• Fr May 08: Fourier spectral methods II. The DFT
• The discrete Fourier transform (DFT) and its application to numerical differentiation, interpolation, and quadrature: supplement to Lects 17-19.
• Mo May 11: Fourier spectral methods III: Relation of DFT to Fourier series; convergence properties vs. function smoothness.
• We May 13: Fourier spectral methods IV: Time-differencing, stability and convergence for advection equation.
• We May 27: Fourier spectral method applied to 2D inviscid fluid flow. Supplemental figs.
• Fr May 29: Finite Element Method introduction and application to advection equation with ICs/BCs.
• Mo May 31: FEM accuracy/stability.
• We Jun 02: Matlab PDE toolbox
• Fr Jun 04: Chebyshev spectral method

Matlab Scripts

Class examples

TimeDifferencingStabilityRegion.m: Plots stability region for amplification eqn for RK2 and RK4 methods; easily adapted to other time-differencing methods.

Nonlinear pendulum d2theta/dt2 = - sin(theta), theta(0) = 1, dtheta/dt = 0 treated as a system of two 1st order ODEs

• pendulum_RK4_plot.m: Plots solution using 4th-order Runge-Kutta for times (0-2)*2*pi and (48-50)*2*pi for a small timestep dt = pi/10 and a larger timestep dt = 4*pi/10 (still only ~1/2 of stability limit).  Uses pendulum_RK4.m.
• pendulum_leapfrog_plot.m: Plots solution to pendulum problem using leapfrog method for times (0-2)*2*pi and (48-50)*2pi , using dt = pi/10 with and without Asselin filtering. Forward-Euler starting step. Uses pendulum_leapfrog.m
• advect_MC.m: Uses MC slope limiter FV method to solve advection equation with c=1 on a periodic domain of length 1 with a square wave initial condition. Plots initial condition and solution after time 1 (when the waveform has advected once around the domain).

Fourier spectral differentiation

• DFT_deriv.m: Takes and plots numerical derivative of a periodic function using Matlab DFT. Uses swell.m and saw.m to generate smooth and sawtooth periodic functions for examples.

  Fourier spectral method on q_t + q_x = 0 in x = [0,1], periodic BCs, 4th order Runge Kutta (RK4) time differencing.

  • spectral_advect_RK4.m: Advects smooth and square-wave periodic functions and plots vs. exact soln.
  • spectral_advect_square_RK4.m: Advects a square wave with various numbers of Fourier modes and plots vs. exact soln.
  • spectral_LW_errconv.m: Error convergence plots vs. dx for a smooth initial condition using Fourier spectral and Lax-Wendroff methods.
  • spectral_MC_squarewave.m: Error convergence plots vs. dx for a square-wave initial condition using Fourier spectral and MC-limited FV methods.

  Fourier spectral method on KdV eqn, periodic BCs, RK4 time differencing.

  • ps_KdV_RK4.m: Advects a KdV soliton and makes plot vs. exact soln. Uses S_KdV.m to calculate pseudospectral approx to spatial derivs for KdV eqn.
  • ps_KdV_2soliton.m: Makes waterfall plot of interacting solitons. Set N=32 instead of N=64 in script to see nonlinear numerical instability. Uses S_KdV.m

  pois_FFT.m: - Fourier spectral method for 2D Poisson eqn. with periodic BC's and RHS = Laplacian of a bivariate Gaussian hump. Makes plot of solution (which recovers the Gaussian hump).

  FS_heat.m: - Fourier spectral method for heat equation u_t = u_xx, 0 < x < 1, with Dirichlet/Neumann BCs, using odd/even extension to a periodic domain. Makes plot of results and error convergence at time 0.0625.

  FS_vortex.m: Fourier pseudospectral method applied to advection of an elliptical vortex in an incompressible 2D fluid. Uses Szeta.m. Figures and technical description in Lecture 23 notes.

  advect_FEM.m: Finite element method on q_t + c*q_x = 0, c=1, 0 < x < 1, 0 < t < 0.5, q(x,0) = 0, q(0,t) = 50*t on a uniform grid with dx = 0.01. Makes plot of solution u(x,0.9). Uses advect_FEMsolve.m, which can also handle a nonuniform grid and a spatially varying c(x).

  chebyshev_advect_RK4.m: Chebyshev spectral method applied to q_t + c(x)*q_x = 0, q(x,0) = 0, q(0,t) = 1. Plots. Uses chebfft.m.

Homework solution scripts

• hw1.m: FTCS method on heat equation du/dt = d2u/dx2 with u(x,0) = 2*x - 1 on 00, with Neumann BCs at x = 0,1. Script plots maxnorm error convergence of solution at time T = 0.032. Uses psiex_hw1.m to calculate the exact solution from a Fourier series.
• hw2p8.m: Implementation and convergence analysis (at t=1) of trapezoidal in time, centered in space differencing of du/dt + du/dx = 0 on periodic domain [0,1], u(x,0) = sin(2*pi*x).
• hw4p5.m: Leapfrog and leapfrog-Asselin centered-difference methods applied to du/dt + du/dx = 0, u(x,0) = 0, u(0,t) = 1.
• hw4xc.m: RK4 centered-difference method applied to du/dt + du/dx = 0, u(x,0) = 0, u(0,t) = 1.
• MC_mid.m: MC driver for midterm problem 4. Uses MC_mid_step.m to march one timestep.
• hw5.m: Driver for solving HW5 problems 2-4 (4 by default; others following instructions in script). Uses MC_step.m and CN_step.m to march one timestep with MC and Crank-Nicholson methods, respectively.
• hw6.m: Driver for solving HW6 (FS method on Burger's equation). Uses hw6_FS_solver.m and S_traffic.m.
• finalp1.m: Driver for solving final prob 1 (FS on nonlinear diffusion equation q_t = (qq_x)_x with q_x=0 at x=0,1 and q(x,0)=sinusoidal pulse centered at x=0.6). Makes two panel plot: (left) space-time plot of solution with N=32 Fourier modes, and (right) Cauchy rms 2N vs. N difference plot at end time T=0.125. Uses nonlindiff_FS_solver.m and S_nonlindiff.m.
• finalp2.m: Driver for solving final prob 2 (FEM with trapezoidal timestepping) on advection equation q_t + (1+9x)q_x = 0 on 0uniform-grid plots for dt=0.01|0.05 and a non-uniform grid plot for dt=0.01. Uses advect_FEMsolve.m.