Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:388

Article

Liu, Yuan¹  Cheng, Yingda^2,3  Chen, Shanqin⁴  Zhang, Yong-Tao⁵ 

[1] Mississippi State Univ, Dept Math & Stat, Mississippi State, MS 39762 USA

[2] Michigan State Univ, Dept Math, E Lansing, MI 48824 USA

[3] Michigan State Univ, Dept Computat Math Sci & Engn, E Lansing, MI 48824 USA

[4] Indiana Univ, Dept Math Sci, South Bend, IN 46615 USA

[5] Univ Notre Dame, Dept Appl & Computat Math & Stat, Notre Dame, IN 46556 USA

关键词: Sparse grid;    Discontinuous Galerkin methods;    Implicit integration factor methods;    Krylov subspace approximation;    Reaction-diffusion equations;   

DOI  :  10.1016/j.jcp.2019.03.021

来源: Elsevier

【 摘 要 】

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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