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 L² 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.

中文翻译：

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非局部粘性守恒律的局部不连续Galerkin方法

本文的目的是研究带有非局部粘性项的标量守恒律的高阶数值近似。粘性项以空间变量的卷积形式给出。借助粘性项的特征，设计了一种半离散局部不连续伽勒金（LDG）方法来求解非局部模型。我们证明了L ²中半离散LDG方法的稳定性和收敛性规范。理论分析表明，对于线性情况，目前的数值方案在最优收敛阶下是稳定的，对于非线性情况，其在次优收敛阶下是稳定的。为了证明我们方案的有效性和准确性，我们使用两个典型的非局部分数粘性项来测试Burgers方程。数值结果表明了线性和非线性情况下空间的收敛阶精度。提供一些数值模拟以显示本数值方案的鲁棒性和有效性。