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An exponential-free Runge--Kutta framework for developing third-order unconditionally energy stable schemes for the Cahn--Hilliard equation

Haifeng Wang, Jingwei Sun, Hong Zhang, Xu Qian, Songhe Song

Abstract

In this work, we develop a class of up to third-order energy-stable schemes for the Cahn--Hilliard equation. Building on Lawson's integrating factor Runge--Kutta method, which is widely used for stiff semilinear equations, we discuss its limitations, such as the inability to preserve the equilibrium state and the oversmoothing of interfacial layers in the solution's profile because of the exponential damping effects. To overcome this drawback, we approximate the exponential term using a class of sophisticated Taylor polynomials, leading to a novel Runge--Kutta framework called exponential-free Runge--Kutta. By incorporating stabilization techniques, we analyze the energy stability of the proposed schemes and demonstrate that they preserve the original energy dissipation without time-step restrictions. Furthermore, we perform an analysis of the linear stability and establish an error estimate in the $\ell^2$ norm. A series of numerical experiments validate the high-order accuracy, mass conservation, and energy dissipation of our schemes.

An exponential-free Runge--Kutta framework for developing third-order unconditionally energy stable schemes for the Cahn--Hilliard equation

Abstract

In this work, we develop a class of up to third-order energy-stable schemes for the Cahn--Hilliard equation. Building on Lawson's integrating factor Runge--Kutta method, which is widely used for stiff semilinear equations, we discuss its limitations, such as the inability to preserve the equilibrium state and the oversmoothing of interfacial layers in the solution's profile because of the exponential damping effects. To overcome this drawback, we approximate the exponential term using a class of sophisticated Taylor polynomials, leading to a novel Runge--Kutta framework called exponential-free Runge--Kutta. By incorporating stabilization techniques, we analyze the energy stability of the proposed schemes and demonstrate that they preserve the original energy dissipation without time-step restrictions. Furthermore, we perform an analysis of the linear stability and establish an error estimate in the norm. A series of numerical experiments validate the high-order accuracy, mass conservation, and energy dissipation of our schemes.

Paper Structure

This paper contains 13 sections, 11 theorems, 103 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Fourier spectral collocation approximations
  3. Exponential-free Runge--Kutta framework
  4. Equilibrium-preserving Taylor-polynomial approximations
  5. Stability analysis
  6. Energy stability of EFRK scheme
  7. Linear stability analysis
  8. Error estimate
  9. Numerical experiments
  10. Test on the preservation of equilibrium state
  11. Convergence tests
  12. Adaptive time stepping
  13. Conclusion

Key Result

Lemma 3.1

higham2008functions If $\phi$ is defined on the spectrum of $M\in \mathbb{C}^{n\times n}$, i.e., the values exist, where $\{\lambda_i\}_{i=1}^n$ are the eigenvalues of $M$, and $n_i$ is the order of the largest Jordan block where $\lambda_i$ appears, we have

Figures (11)

  • Figure 1: Boundaries of the stability regions for EFRK(s,p) with different values of $\theta$.
  • Figure 2: Example \ref{['exm:equilibrium']}: the profiles of the solution $u$ (left column), absolute error (middle column), and energy (right column) computed using the IFRK, EFRK, and splitting schemes.
  • Figure 3: Example \ref{['exm:convtest']}: spatial accuracy tests of the Fourier pseudo-spectral method.
  • Figure 4: Example \ref{['exm:advantage']}: Comparison of solutions computed by EFRK(1,1), EFRK(2,2) and EFRK(3,3) with same time step size $\tau = 10^{-3}$. Parameters: $\epsilon^2 = 0.0025$ and $\kappa = 2.0$.
  • Figure 5: Example \ref{['exm:advantage']}: Evolution of the energy, change in the mass and $\ell^2$-error computed by EFRK(1,1), EFRK(2,2) and EFRK(3,3) with same time step size $\tau = 10^{-3}$. Parameters: $\epsilon^2 = 0.0025$ and $\kappa = 2.0$.
  • ...and 6 more figures

Theorems & Definitions (31)

  • Remark 2.1
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 21 more