论文标题
一种稳定的模拟有限差异方法，用于对流为主导的扩散方程
A Stable Mimetic Finite-Difference Method for Convection-Dominated Diffusion Equations
论文作者
论文摘要
对流扩散方程出现在各种应用中，例如颗粒传输，电磁学和磁流失动力学。即使使用高保真性技术，对对流为主的制度的模拟也尤其具有挑战性，这是由于存在尖锐的边界层和震动而导致解决方案中的跳跃和不连续性，以及数值问题，例如离散化中最大原则的丢失。这些并发症会导致不稳定性，使用传统方法时在数值解决方案中承认大量振荡。与单纯元素元素方法（S. Wu和J. Xu，2020年）的绘制连接，本文使用指数平均的系数开发了模拟有限差异（MFD）离散化，以克服数值解决方案的不稳定性，以作为扩散系数接近零。有限元框架允许对MFD进行透明分析，例如证明适应性和得出误差估计值。给出了数值测试，以确认该方法的稳定性并验证误差估计。
Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques, is particularly challenging due to the presence of sharp boundary layers and shocks causing jumps and discontinuities in the solution, and numerical issues such as loss of the maximum principle in the discretization. These complications cause instabilities, admitting large oscillations in the numerical solution when using traditional methods. Drawing connections to the simplex-averaged finite-element method (S. Wu and J. Xu, 2020), this paper develops a mimetic finite-difference (MFD) discretization using exponentially-averaged coefficients to overcome instability of the numerical solution as the diffusion coefficient approaches zero. The finite-element framework allows for transparent analysis of the MFD, such as proving well-posedness and deriving error estimates. Numerical tests are presented confirming the stability of the method and verifying the error estimates.
