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Krylov integration factor method on sparse grids for high spatial dimension convection-diffusion equations. (English) Zbl 1370.65045

Summary: Krylov implicit integration factor (IIF) methods were developed in [S. Chen and Y.-T. Zhang, J. Comput. Phys. 230, No. 11, 4336–4352 (2011; Zbl 1416.65341)] for solving stiff reaction-diffusion equations on high dimensional unstructured meshes. The methods were further extended to solve stiff advection-diffusion-reaction equations in [T. Jiang and Y.-T. Zhang, J. Comput. Phys. 253, 368–388 (2013; Zbl 1349.65305)]. 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.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35Q84 Fokker-Planck equations
65Y20 Complexity and performance of numerical algorithms

Software:

MATLAB expm
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References:

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