Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A class of refined implicit-explicit Runge-Kutta methods with robust time adaptability and unconditional convergence for the Cahn-Hilliard model

Hong-lin Liao, Tao Tang, Xuping Wang, Tao Zhou

TL;DR

This work develops refined implicit-explicit Runge-Kutta (R-IERK) methods for the Cahn–Hilliard gradient flow that maintain the original energy dissipation law and achieve unconditional $L^2$ convergence without requiring global Lipschitz continuity of the nonlinear bulk. By combining discrete orthogonal convolution kernels, time-space splitting, stabilization, and spectral spatial discretization, the authors construct parameterized second- and third-order schemes whose average dissipation rates are independent of the time-space discretization and adaptivity parameters. A rigorous time-discrete and fully discrete analysis is carried out, establishing uniform bounds on stage solutions, energy stability at all stages, and $L^2$ error estimates with full accuracy under mild regularity assumptions. Numerical experiments validate the theoretical findings, showing robust energy decay and the potential for larger adaptive time steps in long-time CH simulations, with implications for efficient nonlinear stability in gradient-flow computations.

Abstract

One of main obstacles in verifying the energy dissipation laws of implicit-explicit Runge-Kutta (IERK) methods for phase field equations is to establish the uniform boundedness of stage solutions without the global Lipschitz continuity assumption of nonlinear bulk. With the help of discrete orthogonal convolution kernels, an updated time-space splitting technique is developed to establish the uniform boundedness of stage solutions for a refined class of IERK methods in which the associated differentiation matrices and the average dissipation rates are always independent of the time-space discretization meshes. This makes the refined IERK methods highly advantageous in self-adaptive time-stepping procedures as some larger adaptive step-sizes in actual simulations become possible. From the perspective of optimizing the average dissipation rate, we construct some parameterized refined IERK methods up to third-order accuracy, in which the involved diagonally implicit Runge-Kutta methods for the implicit part have an explicit first stage and allow a stage-order of two such that they are not necessarily algebraically stable. Then we are able to establish, for the first time, the original energy dissipation law and the unconditional $L^2$ norm convergence. Extensive numerical tests are presented to support our theory.

A class of refined implicit-explicit Runge-Kutta methods with robust time adaptability and unconditional convergence for the Cahn-Hilliard model

TL;DR

This work develops refined implicit-explicit Runge-Kutta (R-IERK) methods for the Cahn–Hilliard gradient flow that maintain the original energy dissipation law and achieve unconditional convergence without requiring global Lipschitz continuity of the nonlinear bulk. By combining discrete orthogonal convolution kernels, time-space splitting, stabilization, and spectral spatial discretization, the authors construct parameterized second- and third-order schemes whose average dissipation rates are independent of the time-space discretization and adaptivity parameters. A rigorous time-discrete and fully discrete analysis is carried out, establishing uniform bounds on stage solutions, energy stability at all stages, and error estimates with full accuracy under mild regularity assumptions. Numerical experiments validate the theoretical findings, showing robust energy decay and the potential for larger adaptive time steps in long-time CH simulations, with implications for efficient nonlinear stability in gradient-flow computations.

Abstract

One of main obstacles in verifying the energy dissipation laws of implicit-explicit Runge-Kutta (IERK) methods for phase field equations is to establish the uniform boundedness of stage solutions without the global Lipschitz continuity assumption of nonlinear bulk. With the help of discrete orthogonal convolution kernels, an updated time-space splitting technique is developed to establish the uniform boundedness of stage solutions for a refined class of IERK methods in which the associated differentiation matrices and the average dissipation rates are always independent of the time-space discretization meshes. This makes the refined IERK methods highly advantageous in self-adaptive time-stepping procedures as some larger adaptive step-sizes in actual simulations become possible. From the perspective of optimizing the average dissipation rate, we construct some parameterized refined IERK methods up to third-order accuracy, in which the involved diagonally implicit Runge-Kutta methods for the implicit part have an explicit first stage and allow a stage-order of two such that they are not necessarily algebraically stable. Then we are able to establish, for the first time, the original energy dissipation law and the unconditional norm convergence. Extensive numerical tests are presented to support our theory.

Paper Structure

This paper contains 22 sections, 9 theorems, 101 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical and numerical motivations
  3. Theoretical motivation
  4. Numerical motivation
  5. Construction of parameterized R-IERK methods
  6. Second-order R-IERK methods
  7. Third-order R-IERK methods
  8. Energy dissipation laws and $L^2$ norm convergence
  9. Technical lemmas and time-discrete system
  10. Stage regularity of time-discrete solutions
  11. $H_{per}^{2}$ norm boundedness of stage solutions
  12. ($L^2$ norm rough estimate)
  13. ($H_{per}^2$ seminorm rough estimate)
  14. $H_{per}^{m+4}$ norm boundedness of stage solutions
  15. Uniform boundedness of stage solutions and discrete energy law
  16. ...and 7 more sections

Key Result

Lemma 2.1

Assume that the two matrices $D_{\mathrm{E}}$ and $D_{\mathrm{EI}}$ in Def: Differential Matrix D are positive (semi-)definite. If the stage solutions $u_h^{n,\ell}$$(n\ge1,1\le\ell\le s)$ are bounded by ${{K}}_0$ in the maximum norm, and the stabilization parameter $\kappa$ in def: stabilized param for $n\ge1$ and $1\le i\le s_{\mathrm{I}}$. The associated average dissipation rate is nonnegative,

Figures (6)

  • Figure 1: Energy behaviors of R-IERK(1,2) method and IERK(1,2;$\theta$) method \ref{['Scheme: IERK1']}.
  • Figure 2: Solution errors of the two R-IERK methods with different parameters.
  • Figure 3: Energy curves and adaptive time-step sizes of R-IERK(2,4;1) method.
  • Figure 4: Energy behaviors of R-IERK(2,4;$c_2$) method \ref{['scheme: R-IERK-2-4-c2']} and IERK(2,3) method.
  • Figure 5: Energy behaviors of R-IERK(3,6;$\hat{a}_{52}$) method \ref{['scheme: R-IERK-3-6-ah52']} and IERK(3,5) method.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Lemma 2.1
  • Proposition 2.1
  • Example 1
  • Remark 1
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 4 more