An adaptive low-rank splitting approach for the extended Fisher--Kolmogorov equation

TL;DR

This paper introduces a novel rank-adaptive splitting method for the extended Fisher--Kolmogorov equation, improving computational efficiency while preserving key physical properties like energy dissipation.

Contribution

It proposes a new rank-adaptive splitting approach based on a full-rank scheme for the EFK equation, with rigorous convergence and maximum principle proofs.

Findings

The methods are robust and accurate in numerical tests.

They preserve energy dissipation and the discrete maximum principle.

The approach efficiently computes low-rank solutions.

Abstract

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.