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Exponential integrator Fourier Galerkin methods for semilinear parabolic equations

Jianguo Huang, Yuejin Xu

TL;DR

The paper addresses solving semilinear parabolic equations of the form $u_t=\mathcal{D}\Delta u+f(t,u,\nabla u)$ on rectangular domains with periodic boundaries, aiming for high spatial accuracy and efficient computation. It introduces the Exponential Integrator Fourier Galerkin (EIFG) method, which uses Fourier Galerkin discretization in space and an explicit exponential Runge-Kutta scheme in time, yielding a fully discrete scheme with $H^2$-norm error guarantees for two RK stages. Under mild growth and regularity assumptions, the authors prove semi-discrete and fully-discrete error estimates that are independent of the spatial mesh size and time step, for dimensions up to $d\le 3$, and validate these results with comprehensive numerical experiments including convergence tests, Mean Curvature Flow, 3D Burgers, and 3D grain coarsening. The numerical results confirm the predicted rates and demonstrate the method’s efficiency via FFT-based computations. The work provides a solid theoretical and algorithmic framework for higher-order EIFG schemes and potential extensions to localized ETD methods, with practical impact for phase-field and diffusion-reaction systems.

Abstract

In this paper, in order to improve the spatial accuracy, the exponential integrator Fourier Galerkin method (EIFG) is proposed for solving semilinear parabolic equations in rectangular domains. In this proposed method, the spatial discretization is first carried out by the Fourier-based Galerkin approximation, and then the time integration of the resulting semi-discrete system is approximated by the explicit exponential Runge-Kutta approach, which leads to the fully-discrete numerical solution. With certain regularity assumptions on the model problem, error estimate measured in $H^2$-norm is explicitly derived for EIFG method with two RK stages. Several two and three dimensional examples are shown to demonstrate the excellent performance of EIFG method, which are coincident to the theoretical results.

Exponential integrator Fourier Galerkin methods for semilinear parabolic equations

TL;DR

The paper addresses solving semilinear parabolic equations of the form on rectangular domains with periodic boundaries, aiming for high spatial accuracy and efficient computation. It introduces the Exponential Integrator Fourier Galerkin (EIFG) method, which uses Fourier Galerkin discretization in space and an explicit exponential Runge-Kutta scheme in time, yielding a fully discrete scheme with -norm error guarantees for two RK stages. Under mild growth and regularity assumptions, the authors prove semi-discrete and fully-discrete error estimates that are independent of the spatial mesh size and time step, for dimensions up to , and validate these results with comprehensive numerical experiments including convergence tests, Mean Curvature Flow, 3D Burgers, and 3D grain coarsening. The numerical results confirm the predicted rates and demonstrate the method’s efficiency via FFT-based computations. The work provides a solid theoretical and algorithmic framework for higher-order EIFG schemes and potential extensions to localized ETD methods, with practical impact for phase-field and diffusion-reaction systems.

Abstract

In this paper, in order to improve the spatial accuracy, the exponential integrator Fourier Galerkin method (EIFG) is proposed for solving semilinear parabolic equations in rectangular domains. In this proposed method, the spatial discretization is first carried out by the Fourier-based Galerkin approximation, and then the time integration of the resulting semi-discrete system is approximated by the explicit exponential Runge-Kutta approach, which leads to the fully-discrete numerical solution. With certain regularity assumptions on the model problem, error estimate measured in -norm is explicitly derived for EIFG method with two RK stages. Several two and three dimensional examples are shown to demonstrate the excellent performance of EIFG method, which are coincident to the theoretical results.

Paper Structure

This paper contains 14 sections, 12 theorems, 104 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The exponential integrator Fourier Galerkin method
  3. Semi-discretization in space by Fourier Galerkin approximation
  4. Semilinear explicit exponential integrator in time
  5. Convergence analysis for exponential integrator Fourier Galerkin method
  6. Some preliminary lemmas
  7. Semi-discrete error estimate
  8. Fully-discrete error estimate
  9. Numerical experiments and applications
  10. Convergence tests
  11. Mean Curvature Flow
  12. 3D Burgers equations
  13. 3D Grain coarsening simulations
  14. Conclusions

Key Result

Lemma 3.1

HuangJu2023

Figures (5)

  • Figure 1: The shrinking circle at time $t=0,1/4 T, 1/2 T,3/4 T, T$ (from left to right and top to bottom) produced by the EIFG2 scheme for Example \ref{['ex4']} when $d=2$.
  • Figure 2: The shrinking sphere at time $t=0,1/4 T, 1/2 T,3/4 T, T$ (from left to right and top to bottom) produced by the EIFG2 scheme for Example \ref{['ex4']} when $d=3$.
  • Figure 3: The evolutions of $L^2,H^1$-error and $H^2$-error of the numerical solutions produced by the EIFG3 scheme along with the grid sizes in Example \ref{['ex5']}.
  • Figure 4: The simulation process at time $t=0,1/4 T, 1/2 T,3/4 T, T$ (from left to right and top to bottom) produced by the EIFG3 scheme for Example \ref{['ex5']}.
  • Figure 5: The evolutions of supremum norm (left) and energy (right) of the numerical solution produced by EIFG2 scheme for Example \ref{['ex4']}.

Theorems & Definitions (25)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.1
  • Theorem 3.2: Inverse inequality for Fourier space
  • Theorem 3.3
  • Theorem 3.4
  • ...and 15 more