Math 511: Numerical Methods for PDEs
Contents
Catalog Information
Title
Numerical Methods for Partial Differential Equations.
Credit Hours
3
Prerequisite
Math 303 or 347; 410; or equivalents.
Description
Finite difference and finite volume methods for partial differential equations. Stability, consistency, and convergence theory.
Desired Learning Outcomes
Prerequisites
Minimal learning outcomes
Derive finite difference schemes using Taylor series;
Determine the consistency of a difference scheme;
Explain the proper function spaces and discrete norms for grid functions for use in analysis of stability;
Establish the stability of a difference scheme using (1) Heuristic approach (2) Energy method (3) von Neumann method (4) Matrix method;
Recall the CFL condition its relation with stability;
Explain the convergence of the finite difference approximations and its relation with consistency and stability via Lax theorem;
Determine the order of accuracy of a finite difference scheme;
Implement finite difference schemes on computers and perform numerical studies of the stability and convergence properties of the schemes;
Explain the role and the control of numerical diffusion and dispersion in computation ; to determine how numerical phase speed and group velocity may deviate from the theoretical phase speed and group velocity and the numerical techniques to handle such issues;
Recall numerical methods that efficiently handle a multidimensional problem
Recall alternating direction methods that reduce higher dimensional problems into a sequence of one dimensional problems.
Recall the maximum principles for numerical schemes for Laplace equations;
Recall iterative techniques for solving the linear systems resulting from finite element discretization;
Textbooks
Possible textbooks for this course include (but are not limited to):
Randall Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM 2007; ISBN: 0898716292, 978-0898716290
Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd Ed, Cambridge University Press, 2008; ISBN: 0521734908, 978-0521734905
Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009; ISBN: 048646900X, 978-0486469003
Additional topics
Finite element method; Finite volume method; Method of lines
Courses for which this course is prerequisite
Math 303 or 347; 410; or equivalents.
