Spring 2012
http://www.atmos.washington.edu/academics/classes/2012Q2/581/

MWF 2:30-3:20: Loew 216

This class is being offered on-line through EDGE, which provides live streaming of each class and archived lecture videos. You'll need to sign in with your UW netID. If you are an EDGE student, you can also get to this site through the UWEO Moodle portal moodle.extn.washington.edu.

 

Instructor: Prof. Chris Bretherton breth@washington.edu ATG 704, x5-7414 Office hours: MW 1:30-2:20, or by appointment. TA: Mikala Johnson mikalaj@uw.edu Office hours (Gugg 406): Th 1-2; Fr 3:30-4:30 (also via Skype at TA_AMATH586)

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 Texts

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

• Durran, D.R., NEED: Numerical methods for wave equations in geophysical fluid dynamics, 2nd Ed. Springer-Verlag, NEED pp. (We will loosely follows Chapters 1-5; cross-referenced in lecture notes).
• LeVeque, R. J., 2007: Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 339 pp. (used for AMath 585, 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. A GoPost discussion board has been set up for you to discuss homework and other class issues on line. On-campus students shuld submit their homeworks in class, via the class Catalyst dropbox, or to the TA's mailbox in Amath. Please include supporting Matlab scripts. EDGE students, please submit the homework to the Catalyst dropbox. Late homework (after 11:59pm on due date) will be accepted until 5 pm on the second day after the due date. 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. Final assignment will be posted on Wed 30 May, due by 5 pm on Wed 6 June.

Schedule

No class:

• Mo 28 May: Memorial Day (UW holiday)

Instructor on travel; TA will guest-lecture:

• Mo-We 26-28 Mar
• Mo-Fr 16-20 Apr
• Mo 30 Apr

Homework and Exams

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Item	Due Date	Download Solutions
Homework #1	due Mo 9 Apr	HW #1 solutions
Homework #2	due Mo 16 Apr	HW #2 solutions
Homework #3	due We 25 Apr	HW #3 solutions
Homework #4	due We 2 May	HW #4 solutions
Take-home midterm	due Fr 11 May	Midterm solutions
Homework #5	due Fr 18 May	HW #5 solutions
Homework #6	due We 30 May	HW #6 solutions
Take-home final	due We 6 Jun	Final solutions

Lecture Notes and Handouts

Dates subject to revision

• Lecture 1 (Mo Mar 26): Archetypical time-dependent PDEs, well-posedness, finite difference vs. series expansion methods.
• Lecture 2 (We Mar 28): Finite difference operators, local truncation error/order of accuracy, implementation of BCs.
• Lecture 3 (Fr Mar 30): Space-time differencing, accuracy+stability= convergence, energy-integral stability analysis of upwind FDA to advection equation.
• Lecture 4 (Mo Apr 2): Von Neumann stability analysis
• Lecture 5 (We Apr 4): Sufficiency of VN analysis; CFL condition
• Lecture 6 (Fr Apr 6): Aliasing, amplitude/phase error, discrete dispersion relation.
• Lecture 7(Mo Apr 9): Time differencing and amplification equation.
• Lecture 8 (We Apr 11): More single-stage time-differencing methods and their amplification characteristics.
• Lecture 9 (Fr Apr 13): Stability of forward/backward/trapezoidal time differencing for diffusion equation. 4th order Runge-Kutta.
• Lecture 10 (Mo Apr 16): 2,3 order Adams-Bashforth methods and their stability. Intro to leapfrog.
• Lecture 11 (We Apr 18): Leapfrog stability and damping the computational mode; start space differencing generalities
• Lecture 12 (Fr Apr 20): More space differencing generalities for advection equation; inevitability of numerical diffusion/dispersion.
• Lecture 13 (Mo Apr 23): Lax-Wendroff method; conservation laws.
• Lecture 14 (We Apr 25): REA finite volume methods.
• Lecture 14a (Fr Apr 27): REA finite volume methods.
• Lecture 15 (We May 2): Monotonic slope-limiter REA methods.
• Lecture 16 page 1 and pages 2-4 (Fr May 4-Mo May 7): Handling multiple space dimensions, source terms, and nonconstant coefficients. Splitting.
• Lecture 17 (We May 9): Fourier spectral methods I. Basic theory
• Lecture 18 (Fr May 11): 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.
• Lecture 19 (Mo May 14): Fourier spectral methods III: Relation of DFT to Fourier series; convergence properties vs. function smoothness.
• Lecture 20 (We-Fr May 16-18): Fourier spectral methods IV: Time-differencing, stability and convergence for advection equation; pseudospectral method for KdV equation.
• Lecture 21: (Mo May 21) Fourier pseodospectral method applied to 2D inviscid fluid flow.
• Lecture 22: (We May 23) Adaption of Fourier spectral methods to problems with homogeneous Dirichlet or Neumann BCs.
• Lecture 24: (Fr May 25) Finite Element Method introduction and application to advection equation with ICs/BCs.
• Lecture 25: (We May 30) FEM accuracy/stability.
• Lecture 26: (Fri Jun 1) Matlab PDE toolbox

Matlab Scripts

Class Examples

EulerTrapezoidalStabilityRegion.m: Plots stability region and amplification rate |A| for amplification eqn for forward and backward Euler and trapezoidal methods.

RKStabilityRegion.m: Plots stability region (but not amplification rate) 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).

LW_MC_errconv.m: Plot the Lax-Wendroff and MC solutions to advection equation on periodic domain for a sine wave or a square wave advected once around the domain. Left: Nx=8 solution; Right: 2-norm error convergence.

Burgers equation u_t+uu_x = a*u_xx, periodic domain, u(x,0)=1+sin(2*pi*x).

• burgers.m: centered 2nd order space differences, RK4 time difference, unsplit FD. Plots: Inviscid(a=0), T=0.1, Nx = 100 gridpoints, showing initial wave steepening. a=0.01, T=0.5, Nx=25, showing spurious oscillations due to underresolution of shock. Same, with Nx=100, showing better-converged solution, but now timestep dt = 0.005 limited by diffusion.
• burgers_split.m: time-split Burgers equation, using RK4 for advection and Crank-Nicolson for diffusion. Plot shows Nx=100, now with timestep dt = 0.01 twice that of unsplit method because C-N does not diffusively restrict timestep.

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 21 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).

Homework solution scripts

• hw1.m: FTCS method on heat equation du/dt = d2u/dx2 with u(x,0) = 2*x - 1 on 0 < x < 1, t>0, 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. Plot with default parameter settings (dx = 0.05, dt/dx = 0.1, Asselin filtering coefficient gamma = 0.05)
• hw4xc.m: RK4 centered-difference method applied to du/dt + du/dx = 0, u(x,0) = 0, u(0,t) = 1. Plots with max Courant number mu = cmax*dt/dx = 2.71 (stable for RK4) and 4.3 (unstable) .
• 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 0 < x < 1 with q=1 at x=0 and q(x,0)=0, using 20 gridpoints and a boundary element. Modify dt = 0.01|0.05 and grid = 'u'|'n' in script to generated uniform-grid plots for dt=0.01|0.05 and a non-uniform grid plot for dt=0.01. Uses advect_FEMsolve.m.