Difference between revisions of "Math 511: Numerical Methods for PDEs"
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| + | 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 | ||
| + | \begin{itemize} | ||
| + | Heuristic approach Energy method von Neumann | ||
| + | method Matrix method; | ||
| + | \end{itemize} | ||
| + | |||
| + | 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 === | === Textbooks === | ||
Revision as of 12:34, 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
\begin{itemize}
Heuristic approach Energy method von Neumann
method Matrix method; \end{itemize}
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):
