Difference between revisions of "Math 511: Numerical Methods for PDEs"
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| Line 6: | Line 6: | ||
=== Credit Hours === | === Credit Hours === | ||
3 | 3 | ||
| + | |||
| + | === Offered === | ||
| + | W | ||
=== Prerequisite === | === Prerequisite === | ||
| Line 14: | Line 17: | ||
== Desired Learning Outcomes == | == Desired Learning Outcomes == | ||
| + | |||
| + | This course is designed to prepare students to solve mathematical | ||
| + | models represented by initial or boundary value problems involving | ||
| + | partial differential equations that cannot be solved directly | ||
| + | using standard mathematical techniques but are amenable to a | ||
| + | computational approach. Numerical solution of partial differential equations has important applications in many application areas. Students are introduced to the | ||
| + | discretization methodologies, with particular emphasis on the | ||
| + | finite difference method, that allows the construction of | ||
| + | accurate and stable numerical schemes. In depth discussion of | ||
| + | theoretical aspects such as stability analysis and convergence | ||
| + | will be used to enhance the students' understanding of the | ||
| + | numerical methods. Students will also be required to perform some | ||
| + | programming and computation so as to gain experience in | ||
| + | implementing the schemes and to be able to observe the numerical | ||
| + | performance of the various numerical methods. | ||
| + | |||
| + | The course addresses the University goal of developing the skills | ||
| + | of sound thinking, effective communication and quantitative | ||
| + | reasoning. The course also allow students, especially | ||
| + | undergraduate students, to develop some depth and consequently | ||
| + | competence in an important area of applied mathematics. | ||
| + | |||
| + | This course requires knowledge of higher level courses in | ||
| + | mathematics and serves as an introductory graduate | ||
| + | level course to prepare the students to apply the methods learned | ||
| + | in their research projects. | ||
=== Prerequisites === | === Prerequisites === | ||
| + | |||
| + | Understanding of basic theory and properties of solutions of partial differential equations; | ||
| + | |||
| + | Basic programming skill in matlab; | ||
=== Minimal learning outcomes === | === Minimal learning outcomes === | ||
| Line 22: | Line 55: | ||
</div> | </div> | ||
| + | Students are expected to acquire the following knowledge and skills: | ||
| − | Derive finite difference schemes using Taylor series | + | Derive finite difference schemes using Taylor series. |
| − | Determine the consistency of a difference scheme | + | 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 | 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 72: | ||
(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 | ||
| Line 43: | Line 81: | ||
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 | ||
| Line 53: | Line 91: | ||
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 100: | ||
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 difference or finite element discretization. |
=== Textbooks === | === Textbooks === | ||
| Line 71: | Line 109: | ||
Possible textbooks for this course include (but are not limited to): | 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 === | === Additional topics === | ||
| + | Finite element method; Method of lines; Parallel computing | ||
=== Courses for which this course is prerequisite === | === Courses for which this course is prerequisite === | ||
Latest revision as of 10:59, 14 November 2019
Contents
Catalog Information
Title
Numerical Methods for Partial Differential Equations.
Credit Hours
3
Offered
W
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
This course is designed to prepare students to solve mathematical models represented by initial or boundary value problems involving partial differential equations that cannot be solved directly using standard mathematical techniques but are amenable to a computational approach. Numerical solution of partial differential equations has important applications in many application areas. Students are introduced to the discretization methodologies, with particular emphasis on the finite difference method, that allows the construction of accurate and stable numerical schemes. In depth discussion of theoretical aspects such as stability analysis and convergence will be used to enhance the students' understanding of the numerical methods. Students will also be required to perform some programming and computation so as to gain experience in implementing the schemes and to be able to observe the numerical performance of the various numerical methods.
The course addresses the University goal of developing the skills of sound thinking, effective communication and quantitative reasoning. The course also allow students, especially undergraduate students, to develop some depth and consequently competence in an important area of applied mathematics.
This course requires knowledge of higher level courses in mathematics and serves as an introductory graduate level course to prepare the students to apply the methods learned in their research projects.
Prerequisites
Understanding of basic theory and properties of solutions of partial differential equations;
Basic programming skill in matlab;
Minimal learning outcomes
Students are expected to acquire the following knowledge and skills:
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.
