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.
Derive finite volume schemes using flux balance.
Understand how finite volume scheme and finite difference scheme are related.
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 difference or finite element discretization.
Textbooks
Possible textbooks for this course include (but are not limited to):
John Strikwerda, Finite Difference Schemes and Partial Differential Equations, 2nd Ed., SIAM, 2007; ISBN: 089871639X, 978-0898716399
Randall Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM 2007; ISBN: 0898716292, 978-0898716290
K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, 2nd Ed., Cambridge University Press, 2005; ISBN: 0521607930, 978-0521607933
Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd Ed, Cambridge University Press, 2008; ISBN: 0521734908, 978-0521734905
Claes Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, 2009; ISBN: 048646900X, 978-0486469003
J.W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, 2nd Ed., Springer, 2010; ISBN-10: 1441931058, 978-1441931054
Additional topics
Finite element method; Method of lines; Parallel computing
Courses for which this course is prerequisite
Math 303 or 347; 410; or equivalents.
