Advanced Differential Equations: Numerical Solutions to Ordinary and Partial Differential Equations - 625.719

This course presents numerical methods for the solution of both ordinary and partial differential equations. The analytical focus examines concepts of stability and convergence as applied to numerical simulations to differential equations. For solutions to ordinary differential equations topics in Euler’s method and Runge-Kutta methods are considered and analyzed, as well as boundary value problems. For solutions to partial differential equations, both implicit and explicit methods are considered and studied. The majority of consideration will be given to finite difference methods but will include a brief introduction to finite element and discontinuous Galerkin methods. A critical eye will be given toward appropriate discretization and methods, pairing effective techniques to the defined problem. Course work will be divided between analysis and computer implementation through comprehensive projects. Numerical implementations are not required to be in any specific programming language. Some familiarity with programming with a higher-level language (Fortran, MATLAB, Python) will be necessary. Course Notes: This course will complement the development of solutions to differential equations learned in EN.625.717 and EN.625.718, which are largely analytical. EN.625.719 will develop numerical solutions where an analytical solution may be otherwise unavailable. While there is some overlap in the types of differential equations considered, the techniques used to develop solutions are quite different. Similarly, the general concepts of numerical analysis from EN.625.611 are used in this course but applied to a specific application.

Course Prerequisite(s)

625.611 Computational Methods or equivalent graduate-level numerical analysis class. One or more among 625.710, 625.714, 625.717, 625.718, 625.721, 625.725 or equivalent graduate-level mathematics course. (A course in linear algebra or matrix theory will be helpful but not required.)