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; | |
| − | + | ||
Explain the proper function spaces and discrete norms for | Explain the proper function spaces and discrete norms for | ||
| Line 32: | Line 31: | ||
Establish the stability of a difference scheme using | Establish the stability of a difference scheme using | ||
| − | + | (1) Heuristic approach | |
| − | + | (2) Energy method | |
| − | 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 | approximations and its relation with consistency and stability via | ||
Lax theorem; | Lax theorem; | ||
| − | + | Determine the order of accuracy of a finite difference | |
scheme; | scheme; | ||
| − | + | 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 | |
dispersion in computation ; to determine how numerical phase speed | dispersion in computation ; to determine how numerical phase speed | ||
and group velocity may deviate from the theoretical phase speed | and group velocity may deviate from the theoretical phase speed | ||
| Line 57: | Line 56: | ||
issues; | issues; | ||
| − | + | Recall numerical methods that efficiently handle a | |
multidimensional problem | multidimensional problem | ||
| − | + | Recall alternating direction methods that reduce higher | |
dimensional problems into a sequence of one dimensional problems. | dimensional problems into a sequence of one dimensional problems. | ||
| − | + | Recall the maximum principles for numerical schemes for | |
Laplace equations; | Laplace equations; | ||
| − | + | Recall iterative techniques for solving the linear systems | |
resulting from finite element discretization; | resulting from finite element discretization; | ||
Revision as of 12:35, 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):
