The High-Order Variable-Coefficient Explicit-Implicit-Null Method for Diffusion and Dispersion Equations

doi:10.1007/s42967-023-00359-6

Abstract

Abstract: For the high-order diffusion and dispersion equations, the general practice of the explicitimplicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and twodimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CCEIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.

Key words: Diffusion equation, Dispersion equation, Stability, Explicit-implicit-null (EIN) time discretization, Local discontinuous Galerkin (LDG) method

CLC Number:

65M60
65M12

Meiqi Tan, Juan Cheng, Chi-Wang Shu. The High-Order Variable-Coefficient Explicit-Implicit-Null Method for Diffusion and Dispersion Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 115-150.

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