A second-order dynamical low-rank mass-lumped finite element method for the Allen-Cahn equation

TL;DR

This paper introduces a second-order dynamical low-rank mass-lumped finite element method for the Allen-Cahn equation, improving computational efficiency and stability while conserving mass and dissipating energy.

Contribution

It develops a novel second-order low-rank finite element scheme combining Strang splitting and BUG integrator, with enhanced efficiency and stability over existing methods.

Findings

Lower computational complexity than full-rank methods.

Conserves mass up to a truncation tolerance.

Demonstrates stability and robustness in long-time simulations.

Abstract

In this paper, we propose a novel second-order dynamical low-rank mass-lumped finite element method for solving the Allen-Cahn (AC) equation, a semilinear parabolic partial differential equation. The matrix differential equation of the semi-discrete mass-lumped finite element scheme is decomposed into linear and nonlinear components using the second-order Strang splitting method. The linear component is solved analytically within a low-rank manifold, while the nonlinear component is discretized using a second-order augmented basis update & Galerkin (BUG) integrator, in which the $S$-step matrix equation is solved by the explicit 2-stage strong stability-preserving Runge-Kutta method. The algorithm has lower computational complexity than the full-rank mass-lump finite element method. The dynamical low-rank finite element solution is shown to conserve mass up to a truncation tolerance for the conservative Allen-Cahn equation. Meanwhile, the modified energy is dissipative up to a high-order error and is hence stable. Numerical experiments validate the theoretical results. Symmetry-preserving tests highlight the robustness of the proposed method for long-time simulations and demonstrate its superior performance compared to existing methods.