Introduction: In this course, we will emphasize high order accurate numerical methods (e.g. Weighted ENO methods, Discontinous Galerkin Finite Element methods) for solving convection-dominated PDEs (e.g. nonlinear hyperbolic type PDEs, convection-diffusion PDEs) and numerical methods for solving advection-reaction-diffusion PDEs. We will discuss algorithm development, analysis and implementation. Recent advances in high order accuracy numerical methods and their applications in computational physics and computational biology will be discussed. Students will work on course papers and projects.