A monotone combination scheme of diffusion equations on polygonal meshes,ZAMM - Journal of Applied Mathematics and Mechanics

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A monotone combination scheme of diffusion equations on polygonal meshes
ZAMM - Journal of Applied Mathematics and Mechanics ( IF 2.3 ) Pub Date : 2020-03-25 , DOI: 10.1002/zamm.201900320
Fei Zhao ₁ , Zhiqiang Sheng _{2,

3} , Guangwei Yuan ₂

Affiliation

We present a novel monotone scheme which is a combination of linear scheme and nonlinear monotone scheme for solving diffusion problems on general polygonal meshes. It will be called as a combination scheme and consists of two steps. Firstly, a second‐order accurate linear scheme is used to obtain an approximate solution. Secondly, a nonlinear monotone scheme is used to solve the diffusion equation, where the unknowns in the nonlinear coefficient of the nonlinear monotone scheme are taken as the approximate solution of the linear scheme above, i.e., a linearized monotone scheme is obtained. So the combination scheme does not require nonlinear iterations for solving linear diffusion problems, moreover, it benefits from the accuracy and efficiency of the linear scheme, as well as the monotonicity of nonlinear monotone scheme. We also analyze some properties satisfied by the combination scheme, such as conservation, stability, monotonicity and convergence. Numerical results are presented to show the performance of the monotone combination scheme on distorted meshes, especially some numerical comparisons among our combination scheme with some existing linear and nonlinear monotone schemes are given.

中文翻译：

多边形网格上扩散方程的单调组合方案

我们提出了一种新颖的单调方案，该方案是线性方案和非线性单调方案的组合，用于解决一般多边形网格上的扩散问题。它被称为组合方案，包括两个步骤。首先，使用二阶精确线性方案获得近似解。其次，采用非线性单调方案求解扩散方程，将非线性单调方案的非线性系数中的未知数作为上述线性方案的近似解，即得到线性单调方案。因此，组合方案不需要非线性迭代即可解决线性扩散问题，而且受益于线性方案的准确性和效率以及非线性单调方案的单调性。我们还分析了组合方案满足的一些性质，例如守恒性，稳定性，单调性和收敛性。数值结果表明了单调组合方案在变形网格上的性能，特别是我们的组合方案与已有的线性和非线性单调方案之间的一些数值比较。

更新日期：2020-03-25

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