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Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation

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  • Hyun Geun Lee

    (Department of Mathematics, Kwangwoon University, Seoul 01897, Korea)

Abstract

Strong-stability-preserving (SSP) implicit–explicit (IMEX) Runge–Kutta (RK) methods for the Cahn–Hilliard (CH) equation with a polynomial double-well free energy density were presented in a previous work, specifically H. Song’s “Energy SSP-IMEX Runge–Kutta Methods for the Cahn–Hilliard Equation” (2016). A linear convex splitting of the energy for the CH equation with an extra stabilizing term was used and the IMEX technique was combined with the SSP methods. And unconditional strong energy stability was proved only for the first-order methods. Here, we use a nonlinear convex splitting of the energy to remove the condition for the convexity of split energies and give a stability condition for the coefficients of the second-order method to preserve the discrete energy dissipation law. Along with a rigorous proof, numerical experiments are presented to demonstrate the accuracy and unconditional strong energy stability of the second-order method.

Suggested Citation

  • Hyun Geun Lee, 2019. "Stability Condition of the Second-Order SSP-IMEX-RK Method for the Cahn–Hilliard Equation," Mathematics, MDPI, vol. 8(1), pages 1-7, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2019:i:1:p:11-:d:299665
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