论文标题
扩展的Fisher-Kolmogorov方程的自适应低级分裂方法
An adaptive low-rank splitting approach for the extended Fisher--Kolmogorov equation
论文作者
论文摘要
扩展的Fisher-Kolmogorov（EFK）方程已用于描述物理，材料和生物学系统中的某些现象。在本文中，我们提出了一个全等级分裂方案和该方程式的排名自适应分裂方法。我们首先使用有限的差异方法来近似空间衍生物。然后，将所得的半污染系统分为两个硬线性零件和一个非启动的非线性零件。这导致了我们的全级分裂计划。拟议方案的收敛和最大原则被严格证明。基于全等级分裂方案的框架，一种用于获得EFK方程低级别解决方案的等级自适应分裂方法。数值示例表明我们的方法是鲁棒且准确的。他们还可以保留能量耗散和离散最大原则。
The extended Fisher--Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biology systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence and the maximum principle of the proposed scheme are proved rigorously. Based on the frame of the full-rank splitting scheme, a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve energy dissipation and the discrete maximum principle.
