A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization
A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization
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DOI：
10.1016/j.cma.2017.03.020
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发表时间：
2017-06
影响因子：
7.2
通讯作者：
H. Tian;L. Ju;Q. Du
中科院分区：
文献类型：
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作者：
H. Tian;L. Ju;Q. Du
在本文中，我们首先提出了一种非局部对流扩散模型，其中对流项以特殊的迎风方式构建，以便在任何空间维度上都保持质量守恒和极大值原理。建立了所提出的非局部模型的适定性及其与经典局部对流扩散模型的收敛性。然后开发了基于正交的有限差分离散化来数值解决非局部问题，并且它被证明是一致且无条件稳定的。我们进一步证明了数值格式是渐近兼容的，即当δ→0且h→0时，近似解收敛于相应局部问题的精确解。还进行了数值实验来补充理论分析。
In this paper, we first propose a nonlocal convection–diffusion model, in which the convection term is constructed in a special upwind manner so that mass conservation and maximum principle are maintained in any space dimension. The well-posedness of the proposed nonlocal model and its convergence to the classical local convection–diffusion model are established. A quadrature-based finite difference discretization is then developed to numerically solve the nonlocal problem and it is shown to be consistent and unconditionally stable. We further demonstrate that the numerical scheme is asymptotically compatible, that is, the approximate solutions converge to the exact solution of the corresponding local problem when δ→ 0 and h→ 0. Numerical experiments are also performed to complement the theoretical analysis.