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Local discontinuous Galerkin method for a nonlocal viscous conservation laws
Author(s) -
Li Can,
Liu Shuming
Publication year - 2021
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4880
Subject(s) - mathematics , conservation law , nonlinear system , discontinuous galerkin method , numerical analysis , scalar (mathematics) , norm (philosophy) , robustness (evolution) , mathematical analysis , convergence (economics) , order of accuracy , numerical stability , finite element method , law , geometry , physics , biochemistry , chemistry , quantum mechanics , political science , gene , economics , thermodynamics , economic growth
Summary The purpose of this article is to investigate high‐order numerical approximations of scalar conservation laws with nonlocal viscous term. The viscous term is given in the form of convolution in space variable. With the help of the characteristic of viscous term, we design a semidiscrete local discontinuous Galerkin (LDG) method to solve the nonlocal model. We prove stability and convergence of semidiscrete LDG method in L 2 norm. The theoretical analysis reveals that the present numerical scheme is stable with optimal convergence order for the linear case, and it is stable with sub‐optimal convergence order for nonlinear case. To demonstrate the validity and accuracy of our scheme, we test the Burgers equation with two typical nonlocal fractional viscous terms. The numerical results show the convergence order accuracy in space for both linear and nonlinear cases. Some numerical simulations are provided to show the robustness and effectiveness of the present numerical scheme.