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

Yong-Liang Zhao, Xian-Ming Gu

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.

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

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.
Paper Structure (7 sections, 5 theorems, 53 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 7 sections, 5 theorems, 53 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. A full-rank splitting scheme and its convergence
  3. A full-rank splitting scheme
  4. Convergence and discrete maximum principle
  5. A rank-adaptive low-rank splitting approach
  6. Numerical experiments
  7. Concluding remarks

Key Result

Lemma 2.1

(abbaszadeh2020error) Suppose $u \in C^{\infty} (\Omega)$. Then, the nonlinear term $f(u) = u - u^3$ in Eq. eq1.1 satisfies where $L$ is a Lipschitz constant.

Figures (4)

  • Figure 1: The maximum norm (left) and the energy (right) of the numerical solution computed by \ref{['eq2.5']} and \ref{['eq3.5']} for Example 1 with $(M,N) = (128,1024)$.
  • Figure 2: The rank comparison of the numerical solution computed by \ref{['eq2.5']} and \ref{['eq3.5']} for Example 1 with $(M,N) = (128,1024)$.
  • Figure 3: Comparisons of the FRS and ALRS solutions for Example 2 with $M = N = 128$. First two rows: star. Middle two rows: dumbbell. Bottom two rows: torus.
  • Figure 4: The ranks of the numerical solutions computed by \ref{['eq2.5']} and \ref{['eq3.5']} for Example 2 with different initial shapes and $M = N = 128$.

Theorems & Definitions (8)

  • Lemma 2.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof