Difference between revisions of "Math 511: Numerical Methods for PDEs"
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| − | Derive finite difference schemes using Taylor series | + | Derive finite difference schemes using Taylor series. |
| − | Determine the consistency of a difference scheme | + | Determine the consistency of a difference scheme. |
Explain the proper function spaces and discrete norms for | Explain the proper function spaces and discrete norms for | ||
| − | grid functions for use in analysis of stability | + | grid functions for use in analysis of stability. |
Establish the stability of a difference scheme using | Establish the stability of a difference scheme using | ||
| Line 34: | Line 34: | ||
(2) Energy method | (2) Energy method | ||
(3) von Neumann method | (3) von Neumann method | ||
| − | (4) Matrix method | + | (4) Matrix method. |
| − | Recall the CFL condition its relation with stability | + | Recall the CFL condition its relation with stability. |
Explain the convergence of the finite difference | Explain the convergence of the finite difference | ||
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Determine the order of accuracy of a finite difference | Determine the order of accuracy of a finite difference | ||
| − | scheme | + | scheme. |
Implement finite difference schemes on computers and perform | Implement finite difference schemes on computers and perform | ||
numerical studies of the stability and convergence properties of | numerical studies of the stability and convergence properties of | ||
| − | the schemes | + | the schemes. |
Explain the role and the control of numerical diffusion and | Explain the role and the control of numerical diffusion and | ||
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and group velocity may deviate from the theoretical phase speed | and group velocity may deviate from the theoretical phase speed | ||
and group velocity and the numerical techniques to handle such | and group velocity and the numerical techniques to handle such | ||
| − | issues | + | issues. |
Recall numerical methods that efficiently handle a | Recall numerical methods that efficiently handle a | ||
| − | multidimensional problem | + | multidimensional problem. |
Recall alternating direction methods that reduce higher | Recall alternating direction methods that reduce higher | ||
| Line 62: | Line 62: | ||
Recall the maximum principles for numerical schemes for | Recall the maximum principles for numerical schemes for | ||
| − | Laplace equations | + | Laplace equations. |
Recall iterative techniques for solving the linear systems | Recall iterative techniques for solving the linear systems | ||
| − | resulting from finite element discretization | + | resulting from finite element discretization. |
=== Textbooks === | === Textbooks === | ||
Revision as of 12:46, 19 October 2010
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.
