Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Prerequisites

Introduction to Numerical Analysis (18.330), Introduction to Numerical Methods (18.335J)

Description

This course addresses graduate students of all fields who are interested in numerical methods for partial differential equations, with focus on a rigorous mathematical basis. Many modern and efficient approaches are presented, after fundamentals of numerical approximation are established. Of particular focus are a qualitative understanding of the considered partial differential equation, fundamentals of finite difference, finite volume, finite element, and spectral methods, and important concepts such as stability, convergence, and error analysis.

• Problems: advection equation, heat equation, wave equation, Airy equation, convection-diffusion problems, KdV equation, hyperbolic conservation laws, Poisson equation, Stokes problem, Navier-Stokes equations, interface problems.
• Concepts: consistency, stability, convergence, Lax equivalence theorem, error analysis, Fourier approaches, staggered grids, shocks, front propagation, preconditioning, multigrid, Krylov spaces, saddle point problems.
• Methods: finite differences, finite volumes, finite elements, ENO/WENO, spectral methods, projection approaches for incompressible ows, level set methods, particle methods, direct and iterative methods, multigrid.

Textbooks

The course is inspired by the following books, which are all recommended:

LeVeque, Randall J. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2007. ISBN: 9780898716290.

———. Finite Volume Methods for Hyperbolic Problems. Cambridge texts in applied mathematics. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521009249.

Fletcher, C. A. J. Computational Techniques for Fluid Dynamics. Fundamental and General Techniques Volume I. Springer series in computational physics. New York, NY: Springer-Verlag, 1996. ISBN: 9783540530589.

———. Computational Techniques for Fluid Dynamics. Specific Techniques for Different Flow Categories Volume II. Springer series in computational physics. New York, NY: Springer-Verlag, 1991. ISBN: 9783540536017.

Canuto, Claudio S., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics. New York, NY: Springer-Verlag, 2007. ISBN: 9783540307273.

Trefethen, Lloyd N. Spectral Methods in MATLAB (Software, Environments, Tools). Philadelphia, PA: Society for Industrial and Applied Mathematics, 2001. ISBN: 9780898714654.

Evans, Lawrence C. Partial Differential Equations. Vol. 19. Graduate studies in mathematics. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.

Grading

Assignments

On average, there will be one problem set assigned every two weeks, with exercises on theory and programming (50% of work load). Most homework problems involve programming. This course requires and encourages flexibility with programming tools. MATLAB is the main language used for small to medium programs and all visualization purposes. In principle programming can be done in any language. However, the use of special software packages is not allowed, unless specifically required. You are also not allowed to consult solutions from previous years.

Course Projects

The course project is to be worked on over the whole term, starting in the second week. The project may relate to your thesis work, but the project work must be unique to this course.

Exams

There will be no exams.