High Order Numerical Methods for Solving Partial Differential Equations
In this project, we develop efficient and robust high order accuracy numerical methods for solving partial differential equations (PDEs) including hyperbolic conservation laws and convection dominated equations, Hamilton-Jacobi equations, and stiff advection-reaction-diffusion equations, etc. The numerical methods we study include weighted essentially non-oscillatory (WENO) finite difference / finite volume methods, large time-stepping schemes (e.g., high order integration factor schemes and their high dimensional implementation, and other exponential integrators), fast and robust iterative methods for steady state problems (e.g., fast sweeping methods), sparse grid methods for solving high dimensional problems, discontinuous Galerkin (DG) finite element methods, and high order accuracy numerical methods on unstructured meshes for complex domain geometries, etc.