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Energetic Spectral-Element Time Marching Methods for Phase-Field Nonlinear Gradient Systems

Shiqin Liu, Haijun Yu

TL;DR

This work presents two high-order, energy-stable spectral-element time marching schemes (an implicit nonlinear and a semi-implicit linear variant) for nonlinear gradient systems, demonstrated on the Allen–Cahn equation. The schemes are built on an energetic weak formulation that preserves mass conservation (when applicable) and energy dissipation at the discrete level, with superconvergence for the fully implicit method. A semi-implicit version uses a stabilized extrapolation and constant-coefficient linear systems, achieving efficiency, while a Picard-like iteration for the implicit scheme recovers higher-order accuracy. Numerical tests show strong energy stability, mass conservation in the conservative case, and competitive performance relative to existing fourth-order methods, with excellent scalability via a diagonalization solver. The methods are positioned as broadly applicable to large-scale nonlinear dynamical systems beyond the phase-field setting, though extensions to $H^{-1}$ gradient problems require further stabilization work.

Abstract

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen--Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen--Cahn equation, we also apply the method to a conservative Allen--Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen--Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Energetic Spectral-Element Time Marching Methods for Phase-Field Nonlinear Gradient Systems

TL;DR

This work presents two high-order, energy-stable spectral-element time marching schemes (an implicit nonlinear and a semi-implicit linear variant) for nonlinear gradient systems, demonstrated on the Allen–Cahn equation. The schemes are built on an energetic weak formulation that preserves mass conservation (when applicable) and energy dissipation at the discrete level, with superconvergence for the fully implicit method. A semi-implicit version uses a stabilized extrapolation and constant-coefficient linear systems, achieving efficiency, while a Picard-like iteration for the implicit scheme recovers higher-order accuracy. Numerical tests show strong energy stability, mass conservation in the conservative case, and competitive performance relative to existing fourth-order methods, with excellent scalability via a diagonalization solver. The methods are positioned as broadly applicable to large-scale nonlinear dynamical systems beyond the phase-field setting, though extensions to gradient problems require further stabilization work.

Abstract

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen--Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen--Cahn equation, we also apply the method to a conservative Allen--Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen--Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Paper Structure

This paper contains 18 sections, 10 theorems, 139 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The energetic spectral-element time marching (ESET) schemes
  3. Galerkin approximation on a single element
  4. Spectral-element Galerkin approximation: the implicit scheme
  5. Spectral-element Galerkin approximation: the semi-implicit linear scheme
  6. Implementation
  7. Stability and convergence analysis
  8. Some preliminaries
  9. Energy stability of the semi-implicit linear ESET method
  10. Convergence analysis
  11. A standard error estimate for the implicit ESET scheme
  12. Error estimate for the semi-implicit linear ESET scheme
  13. The semi-implicit iterative solver for the implicit scheme and superconvergence
  14. Convergence of the iteration
  15. Uniqueness of solution to the implicit ESET scheme
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose For $\tau = |t_n - t_{n-1}| \le \varepsilon/L$, equation eq:SEGal has a unique solution $h^n$, and it satisfies the discrete energy law

Figures (10)

  • Figure 1: Testing ESET with Dirichlet(left) and Neumann(right) boundary condition, respectively. The scheme parameters used: $M = 255, N=3$, $\tau = 0.01$, $S=0$.
  • Figure 2: Accuracy and efficiency of ESET, IMEX4, and ETDRK4 schemes. The common parameters used: $\varepsilon=0.05$, $M = 350$, $S=0$.
  • Figure 3: The energy dissipation of ESET31 scheme \ref{['eq:SEGalLinear']} with different stabilization constant $S$ for $\varepsilon=0.08$ and $M=350$.
  • Figure 4: Effects on stability and accuracy of the stabilization constant and cut-off operation to maintain maximum principle. $M=350$. In this figure, the lines break at a time step size when the corresponding numerical scheme encounters blow-ups.
  • Figure 5: Comparison of accuracy and efficiency between the diagonalization method and direct sparse solver of the ESET42 scheme with $M=350, \varepsilon=0.08, T=1.2$.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • ...and 11 more