Numerical Methods For Partial Differential Equations

Lecturer: Dr Tim Warburton

Books

Finite Element Method: Linear Static and Dynamic Finite Element Analysis,
Thomas J. R. Hughes
(Dover Publications)

Finite Volume Methods for Hyperbolic Problems,
by Randall J. LeVeque, D. G. Crighton (Series Editor)
(Cambridge Texts in Applied Mathematics)

Time Dependent Problems and Difference Methods
by Bertil Gustafsson, Heinz-Otto Kreiss, Joseph Oliger
(Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)


Links to some useful Unix tutorials
 
Links to some quick introductory guides for the text editor Emacs:

Week 1


Week 2

Lecture 2: stability of Euler-Forward and introducing AB2 time stepping

Lecture 3: stability of AB2, AB3. Method order accuracy. Consistency. Convergence of linear multistep time stepping. Homework 1 due 01/27/05 beginning of class.

Week 3

Lecture 4: one-step time stepping schemes. Runge-Kutta methods


Lecture 5: summary of stability/consistency and introduction to difference formulae for derivatives (homework)

Week 4

Lecture 6: analyzing the spectrum of some finite difference operators (introduction to numerical dispersion and dissipation)

Lecture 7: demonstrations of the effects of numerical dispersion and dissipation. Homework 3 due 02/10/05


Week 5

Spring 2006 Homework 3  (ppt)(pdf)

Lecture 8: overview of convergence and accuracy for finite difference schemes, brief discussion of boundary conditions via the energy method
(see Lecture 7 for correction to Q1f initial condition)

Lecture 9: full description of solutions for hw3

Week 6

Lecture 10: Basic finite volume method

Week 7

Lecture 11: higher-resolution finite volume methods, basic limiter.


Lecture 12: flux limiter functions, Sweby TVD stability diagrams, Harten Theorem.
(homework 4, due 03/03/2005).

Week 8

Lecture 13: scalar nonlinear conservation laws (MIT notes).

Lecture 14: finite volume methods for scalar nonlinear conservation laws, conservation property, Lax-Wendroff theorem (MIT notes). No homework this week, have good spring break.

Week 9

Spring break

Week 10

Lecture 15: 2D finite-volume on triangle meshes. Project. Topology and geometry of triangle meshes, computing connectivity.


Lecture 16. Project 1: background material

Project 1: Matlab code example

Week 11

Lecture 17: Interpolation on the triangle (Proriol's orthonormal polynomial basis). Integrating and differentiating interpolants on the triangle. Brief derivations of discontinuous Galerkin for the advection equation.


To run under *nix:
> unzip umSYMB.zip
> cd umSYMB
> matlab
>> umSYMBDEMO2d

To compute the (n,m)'th orthonormal Proriol basis function expansion in Matlab:


>> umSYMBPKDO2d(n,m)


Examples:


>> umSYMBPKDO2d(1,0)


ans =


1/2*3^(1/2)*(1+2*r+s)


>> umSYMBPKDO2d(3,2)


ans =


1/32*21^(1/2)*(s^2+8*s+10*r*s+10*r+1+10*r^2)*(1+2*r+s)*(19+70*s+55*s^2)

Lecture 18: Building blocks for discontinuous Galerkin on a triangle grid.

Week 13

Project 2: Final project description


Lecture 19: Notes on a basic, Bubnov-Galerkin, linear finite element method in 1D.

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