The extended Fisher–Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biological 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 of the proposed scheme is proved rigorously. Based on the frame of the full-rank splitting scheme, we design 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 the energy dissipation.

中文翻译：

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扩展 Fisher-Kolmogorov 方程的自适应低秩分裂方法

扩展的费希尔-柯尔莫哥洛夫 (EFK) 方程已被用来描述物理、材料和生物系统中的一些现象。在本文中，我们提出了该方程的全秩分裂方案和秩自适应分裂方法。我们首先使用有限差分方法来近似空间导数。然后，将所得的半离散系统分为两个刚性线性部分和一个非刚性非线性部分。这导致了我们的全等级分裂方案。严格证明了所提出方案的收敛性。基于全秩分裂方案的框架，我们设计了一种秩自适应分裂方法来获得EFK方程的低秩解。数值例子表明我们的方法是稳健且准确的。它们还可以保留能量耗散。