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

Jun Yang, Nianyu Yi, Peimeng Yin

TL;DR

The paper develops a second-order dynamical low-rank mass-lumped finite element method (DLR-MLFEM) to solve the Allen-Cahn equation by splitting the semi-discrete matrix equation into linear and nonlinear parts. An exact low-rank integrator handles the linear subproblem, while a second-order augmented BUG-based integrator advances the nonlinear part, yielding a method with complexity O((m+n)r^4) that preserves mass up to truncation and dissipates a modified energy with high-order accuracy. The framework extends to both classical and conservative Allen-Cahn formulations, demonstrates mass conservation, energy dissipation, and symmetry-preserving long-time behavior, and shows competitive accuracy and robustness in numerical experiments. This work advances efficient, structure-preserving simulations of gradient-flow PDEs by integrating DLRA with mass-lumped FEM and high-order splitting, enabling scalable simulations on large 2D/3D problems.

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.

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

TL;DR

The paper develops a second-order dynamical low-rank mass-lumped finite element method (DLR-MLFEM) to solve the Allen-Cahn equation by splitting the semi-discrete matrix equation into linear and nonlinear parts. An exact low-rank integrator handles the linear subproblem, while a second-order augmented BUG-based integrator advances the nonlinear part, yielding a method with complexity O((m+n)r^4) that preserves mass up to truncation and dissipates a modified energy with high-order accuracy. The framework extends to both classical and conservative Allen-Cahn formulations, demonstrates mass conservation, energy dissipation, and symmetry-preserving long-time behavior, and shows competitive accuracy and robustness in numerical experiments. This work advances efficient, structure-preserving simulations of gradient-flow PDEs by integrating DLRA with mass-lumped FEM and high-order splitting, enabling scalable simulations on large 2D/3D problems.

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 -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.
Paper Structure (20 sections, 13 theorems, 138 equations, 12 figures)

This paper contains 20 sections, 13 theorems, 138 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Matrix differential equation
  3. 1D semi-discrete finite element scheme
  4. Matrix differential equation in 2D
  5. Second-order FR-MLFEM
  6. Dynamical low-rank mass-lumped finite element method
  7. An exact low-rank integrator for the linear problem \ref{['2.20a']}
  8. DLRA for nonlinear problem \ref{['2.20b']}
  9. Low complexity strategy for the nonlinear and nonlocal terms
  10. Second-order integrator
  11. Second-order DLR-MLFEM
  12. Numerical experiment
  13. Convergence test
  14. Modified energy
  15. Mass-conservation
  16. ...and 5 more sections

Key Result

Lemma 2.1

\newlabellem2.20 For any $z_h, w_h \in Q_h^k$, let $Z, W \in \mathbb{R}^{m \times n}$ be their coefficient matrices, respectively. Then, and for any integer $s>0$, Moreover, where the matrices $L_x = -M^{-1}_x A_x \in \mathbb{R}^{m\times m}, \ L_y = -M^{-1}_y A_y \in \mathbb{R}^{n \times n}$.

Figures (12)

  • Figure 1: Spatial accuracy tests of the DLR-MLFEM at $T=1$.
  • Figure 2: Temporal accuracy tests of the DLR-MLFEM at $T=1$.
  • Figure 3: Evolution of Energy and modified energy calculated by different methods; Left: FR-MLFEM. Righe: adaptive DLR-MLFEM. Parameters: $N_x = N_y = 129$, tolerance $\eta = 0.01\|\Sigma\|_2$.
  • Figure 4: Snapshots of solutions of AC equation computed using FR-MLFEM (first row) and adaptive DLR-MLFEM (second row). Parameters: $N_x = N_y = 256$, $\tau = 0.5$, $\eta = 0.01\|\Sigma\|_{2}$.
  • Figure 5: Snapshots of solutions of AC equation with RSLM, computed using FR-MLFEM(first row) and adaptive DLR-MLFEM(second row). Parameters: $N_x = N_y = 256, \tau = 0.5, \eta = 0.001\|\Sigma\|_{2}.$
  • ...and 7 more figures

Theorems & Definitions (38)

  • Remark 2.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Proof 1
  • Corollary 2.1
  • Corollary 2.2
  • Remark 2.2
  • Lemma 2.2
  • Lemma 2.3
  • ...and 28 more