NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

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 1 Syllabus  (syllabus: doc tentative)      (syllabus: pdf tentative) Lecture 1 (lecture ppt)     (lecture pdf)  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.m           ABimagnu.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. (lecture ppt)                        (lecture pdf)  exactsolution.m (needed for scripts below) leftdifference1.m (unstable)                           rightdifference1.m centraldifference2.m             centraldifference4.m             centraldifferenc6.m laxfriedrichs.m                    testrig.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.m fvsolver.m                     minmod.m Lecture 12: flux limiter functions, Sweby TVD stability diagrams, Harten Theorem. (lecture ppt)                (lecture pdf)                      fluxlimiter.m fluxlimiterexact.m                  minmod.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 9 Spring 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.m umSLOWCONNECT.m              umFASTCONNECT.m          MeshReader.zip WUM_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).