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

TL;DR

This paper introduces Krylov IIF discontinuous Galerkin methods on sparse grids for efficiently solving high-dimensional reaction-diffusion PDEs, reducing computational costs and handling stiffness effectively.

Contribution

It develops a novel combination of DG spatial discretization with multiwavelet bases and Krylov IIF time integration on sparse grids for high-dimensional reaction-diffusion equations.

Findings

Achieves significant reduction in degrees of freedom using multiwavelet bases.

Demonstrates stability and high accuracy in numerical examples.

Enables large time step computations for stiff reaction-diffusion systems.

Abstract

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.