Spring 2005
http://www.atmos.washington.edu/2005Q2/581/

MWF 2:30-3:20: ATG 310c

 

Instructor: Prof. Chris Bretherton breth@atmos.washington.edu ATG 710, x5-7414 Office hours: TuTh 1:30-2:20, or by appointment.

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.

Prerequisites

Notes and Recommended Text

No text will be required and class notes will be handed out. The presentation will loosely follow parts of Chapters 1-5 of:

• Durran, D.R., 1999: Numerical methods for wave equations in geophysical fluid dynamics. Springer-Verlag, 465 pp. ($65, cheaper on Amazon)

Syllabus

Grading

• Homework (60%), posted to class web page and assigned on a quasi-weekly basis. 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. Late homework (after 5pm on due date, but not more than three school days late) accepted by arrangement with instructor. 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.

Schedule

No class:

• Mo 4 Apr (CB talks at Caltech)
• We 11 May (CB at EPIC meeting, APL)
• Mo 16 May-Fr 27 May (CB in Greece for pan-GCSS conference)
• Mo 30 May: Memorial Day (vacation)

Makeup classes: Every Thursday through 5 May, and Th 2 June at 2:30-3:20 in ATG 310c

Homework and Exams

Matlab Scripts and Handouts

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 (18-20)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 (49-50)*2pi , using dt =  pi/10 with and without Asselin filtering.  Forward-Euler starting step. Uses pendulum_leapfrog.m

Stiff ODE example: stiff.m: Compares exact solution to du/dt = -lambda(u - sin(t)), u(0) = 1 with trapezoidal and backward-Euler methods.

Fourier spectral differentiation

• DFT-based differentiation notes
• DFT_deriv.m: Taking a 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 a smooth periodic function and overplots vs. exact soln.
  • spectral_advect_square_RK4.m: Advects a square wave and overplots vs. exact soln.

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

  • ps_KdV_RK4.m: Advects a KdV soliton and plots vs. exact soln. Uses S_KdV.m
  • ps_KdV_2soliton.m: Interacting solitons. Uses S_KdV.m
  • S_KdV.m: Calculates pseudospectral approx to spatial derivs for KdV eqn.

  pois_FFT.m: - Fourier spectral method for 2D Poisson eqn. with periodic BC's.

  PDE toolbox example using heat equation on a semicircular domain.

• hw1g.m: FTCS method on heat equation du/dt = d2u/dx2 with u(x,0) = 1 - cos(2*pi*x) on 00, with Neumann BCs at x = 0,1. Script plots maxnorm error convergence of solution at time T = 0.016.
• 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).
• hw3p1bc.m: Convergence analysis of numerical solutions to a stiff system of 2 ODEs using backward Euler and BDF2.
• hw3p2c.m: Leapfrog and leapfrog-Asselin centered-difference methods applied to du/dt + du/dx = 0, u(x,0) = 0, u(0,t) = 1.
• hw4p234.m: Traffic-flow problem dq/dt + (1-2q)dq/dx = 0, on x = [0,1], q(x,0)= 0, q(0,t) = 0.12sin^2(pi*t). Solution at right boundary plotted for exact, leapfrog-Asselin, L-W and MC methods. Uses hw4_fv.m
• hw5_soln.m:  Fourier pseudospectral method applied to advection of an elliptical vortex in an incompressible 2D fluid.  
• finalp1.m: FEM applied to advection equation dq/dt + dq/dx = 0 on x = [0,1], q(x,0)=0, q(0,t) = sin(50t), with solution plotted at time 0.9 for a uniform and stretched grid in x. Uses finalp1_FEMsolve.m