Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

2019/03/28 by Haijin Wang, Wang, Haijin, Qiang Zhang +5
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1903.11859

openalex publication_date 2019/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper we discuss the local discontinuous Galerkin methods coupled with two specific explicit-implicit-null time discretizations for solving one-dimensional nonlinear diffusion problems U_t=(a(U)U_x)_x. The basic idea is to add and subtract two equal terms a₀ U_xx on the right hand side of the partial differential equation, then to treat the term a₀ U_xx implicitly and the other terms (a(U)U_x)_x-a₀ U_xx explicitly. We give stability analysis for the method on a simplified model by the aid of energy analysis, which gives a guidance for the choice of a₀, i.e, a₀ ≥ max\a(u)\/2 to ensure the unconditional stability of the first order and second order schemes. The optimal error estimate is also derived for the simplified model, and numerical experiments are given to demonstrate the stability, accuracy and performance of the schemes for nonlinear diffusion equations.

Citations

• A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier–Stokes Equations

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