Numerical Methods for PDEs

Catalog description

The objective of the course is to provide training in fundamentals of numerical analysis at the PhD level. This course does not explore methods tailored to a specific physics subdomain. Instead, general ideas and systematic procedures for construction and analysis of numerical methods are introduced, which can be applied to diverse disciplines of computational science seeking numerical solution of complex differential equations. The course begins with the techniques for numerical differentiation/integration and solution of system of ODEs, which later is integrated into techniques for solution of PDEs of various types (hyperbolic, parabolic, and elliptic ones). Measures of stability and accuracy are presented, with emphasis on preserving symmetries of differential operators to preclude unphysical solution growth or decay. Spectral methods based on Sturm-Liouville eigenfunctions are covered. Utility and limitation of the widely used methods for computation of broadband phenomena are illustrated. Students will have first-hand experience writing their own computer programs, and also come to view critically the numerical output generated by a computer. Background on linear algebra and ordinary/partial differential equations at the level of ENM 5100 and basic MATLAB experience. Undergraduates require instructor permission to enroll.