Math 511: Numerical Methods for PDEs
Numerical Methods for Partial Differential Equations.
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.
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.
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
Finite element method; Method of lines; Parallel computing
Courses for which this course is prerequisite
Math 303 or 347; 410; or equivalents.