多尺度介质中的非局部输运方程。建模、记忆和离散化
Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations
摘要：
本文考虑了一类具有记忆效应的对流-扩散方程。这些方程是非均匀介质中线性输运方程的均匀化或尺度放大的结果，在许多应用中起着重要的作用。首先，我们提出了一个记忆技术，这些方程。证明了具有记忆效应的对流-扩散方程可以写成标准对流-扩散-反应方程组。这允许去除记忆项并简化计算。我们考虑记忆方程和微尺度方程之间的关系，其中不包含记忆项。我们注意到，记忆方程不同于微观尺度方程，并构成一个宏观模型。接下来，我们考虑隐式和部分显式方法。后者是针对具有高对比度特性的多尺度介质中的问题而引入的。由于高对比度，显式方法是限制性的，并且需要非常小的时间步长（比例为对比度的倒数）。我们表明，通过适当地分解空间，我们可以对待只有几个自由度隐式和其余的自由度明确。我们提出了一个稳定性分析。数值结果证实了我们的理论研究结果的部分显式计划适用于记忆系统的方程。
英文摘要：
In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection-diffusion-reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and micro-scale equations, which do not contain memory terms. We note that dememorized equations differ from micro-scale equations and constitute a macroscopic model. Next, we consider both implicit and partially explicit methods. The latter is introduced for problems in multiscale media with high-contrast properties. Because of high contrast, explicit methods are restrictive and require time steps that are very small (scales as the inverse of the contrast). We show that, by appropriately decomposing the space, we can treat only a few degrees of freedom implicitly and the remaining degrees of freedom explicitly. We present a stability analysis. Numerical results are presented that confirm our theoretical findings of partially explicit schemes applied to dememorized systems of equations.
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