PARTIAL DIFFERENTIAL EQUATIONS

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS CAAM 452 Lecturer: Dr Tim Warburton

	Week 1 Syllabus and Lecture 1(syllabus: doc tentative)(syllabus: pdf tentative)(lecture ppt)(lecture pdf) 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) Free online: Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations Lloyd N. Trefethen Links to some useful unix tutorials unixtools.com tutorial CS at U. Washington Unix intro Unix notes from ITEP in Russia Notes from U. Northern Iowa CS at U. Tennessee Unix notes DECF at Berkeley Unix notes Links to some quick introductory guides for the text editor Emacs: Emacs notes by Paul Seamon Emacs notes by Bob Rogers Emacs notes at CS Utah Computer support Portland State emacs notes
	Week 2 Lecture 2: stability of Euler-Forward and introducing AB2 time stepping(lecture ppt)(lecture pdf) EulerForwardODE.m 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.(lecture ppt)(lecture pdf) ABabstab1.mABimagnu.m
	Week 3 Lecture 4: one-step time stepping schemes. Runge-Kutta methods(lecture ppt)(lecture pdf) RKabstab.m [Optional homework/report template for latex] (you may also use your own similar style sheet, or a word processor) Lecture 5: summary of stability/consistency and introduction to difference formulae for derivatives (homework)(lecture ppt)(lecture pdf) cashkarp.m (coefficients for the Cash-Karp 6 stage, 5th order Runge-Kutta with embedded 4th order solution.
	Week 4 Lecture 6: analyzing the spectrum of some finite difference operators (introduction to numerical dispersion and dissipation)(draft lecture ppt)(draft lecture pdf) plotmultiplies.m Lecture 7: demonstrations of the effects of numerical dispersion and dissipation. Homework 3 due 02/10/05(lecture ppt)(lecture pdf) exactsolution.m (needed for scripts below) leftdifference1.m (unstable)rightdifference1.m centraldifference2.mcentraldifference4.mcentraldifferenc6.m laxfriedrichs.mtestrig.m
	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)(draft lecture ppt)(draft lecture pdf) Lecture 9: full description of solutions for hw3 (lecture ppt)(lecture pdf)
	Week 6 Lecture 10: Basic finite volume method(draft lecture ppt)(not ready lecture pdf)
	Week 7 Lecture 11: higher-resolution finite volume methods, basic limiter.(lecture ppt)(lecture pdf) fvexact.mfvsolver.mminmod.m Lecture 12: flux limiter functions, Sweby TVD stability diagrams, Harten Theorem. (homework 4, due 03/03/2005).(lecture ppt)(lecture pdf) fluxlimiter.mfluxlimiterexact.mminmod.m sweby.m
	Week 8 Lecture 13: scalar nonlinear conservation laws (MIT notes).(MIT online course notes Aeronautics and Astronautics )(lecture slides)(lecture 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.(MIT online course notes Aeronautics and Astronautics )(lecture slides)(lecture notes)
	Week 9Spring break
	Week 10 Lecture 15: 2D finite-volume on triangle meshes. Project. Topology and geometry of triangle meshes, computing connectivity.(corrected lecture ppt)(corrected lecture pdf) umCONNECT.mumSLOWCONNECT.mumFASTCONNECT.mMeshReader.zipWUM_v5.zip (access password needed) Lecture 16. Project 1: background material(scanned lecture pdf)Project 1: Matlab code example(Code directory)(Example Matlab html output)
	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.(scanned lecture pdf) umSYMB.zip demonstrating polynomial expansions for Proriol basis functions (analytic integration and differentiation using Matlab's built in symbolic package). 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.(scanned lecture pdf) Links to papers on computing and using high order nodal interpolation sets for the triangle: (Chen and Babuska's optimized interpolation points for the triangle) (Taylor, Wingate and Vincent's algorithms for computing the Fekete nodes on the triangle) (Presentation by Taylor, Wingate & Bos on trying to find good interpolation points which also serve as a good quadrature) (Hesthaven's paper on computing the electrostatic prinicpal nodes for the triangle) (Heinrich's paper on improved interpolation points for the triangle) (Paper outlining techniques described in class to handle high order nodal triangles) (Nice paper on using nodal triangles with curvature) (Lecture notes on DG by Bernardo Cockburn @ UMN) (Matlab Gauss quadrature routine by Greg von Winckel on Matlab Central)
	Week 13 Project 2: Final project description(pdf)(NodesCAAM452.zip containing node locations, derivative matrices and lift matrices) Lecture 19: Notes on a basic, Bubnov-Galerkin, linear finite element method in 1D.(low quality scanned lecture pdf)
	Week 14 Lecture 13: Finite elements in 2D.