Krylov Integration Factor Method on Sparse Grids for High Spatial Dimension Convection–Diffusion Equations

Abstract

Krylov implicit integration factor (IIF) methods were developed in Chen and Zhang (J Comput Phys 230:4336---4352, 2011) for solving stiff reaction---diffusion equations on high dimensional unstructured meshes. The methods were further extended to solve stiff advection---diffusion---reaction equations in Jiang and Zhang (J Comput Phys 253:368---388, 2013). Recently we studied the computational power of Krylov subspace approximations on dealing with high dimensional problems. It was shown that the Krylov integration factor methods have linear computational complexity and are especially efficient for high dimensional convection---diffusion problems with anisotropic diffusions. In this paper, we combine the Krylov integration factor methods with sparse grid combination techniques and solve high spatial dimension convection---diffusion equations such as Fokker---Planck equations on sparse grids. Numerical examples are presented to show that significant computational times are saved by applying the Krylov integration factor methods on sparse grids.