Asymptotic Methods in Computational Engineering

This course is not a standard teaching of asymptotic methods as thought in the applied math programs. Nor does it involve such elaborate algebra and analytical derivations. Instead, the class relies on students' numerical programming skills and introduces improvements on numerical methods using standard asymptotic and scaling ideas. The main objective of the course is to bring physical insight into numerical programming. The majority of the problems to be explored involve one- and two-dimensional transient partial differential equations inspired by thermal-fluid and transport engineering applications. Topics include: 1-Review of numerical discretization and numerical stability, 2-Implicit versus explicit methods, 3-Introduction to regular and singular perturbation problems, 4-Method of matched asymptotic expansions, 5-Stationary thin interfaces: boundary layers, Debye layers, 6-Moving thin interfaces: shocks, phase-interfaces, 7-Reaction-diffusion problems, 8-Directional equilibrium and lubrication theory.