In this paper, we present an efficient and energy-stable scheme based on the Crank–Nicolson formula for the modified phase-field crystal equation with a strong nonlinear vacancy potential. In the scheme, the nonlinear terms (the first derivatives of the double-well and vacancy potentials) are treated explicitly, which makes the scheme efficient, and the energy stability is guaranteed by assuming that the second derivatives of the double-well and vacancy potentials are each bounded and by adding two second-order stabilization terms. In particular, by bounding the second derivatives of the double-well and vacancy potentials, respectively, we can choose the stabilization parameters independently of the vacancy parameter. As a result, the convergence constant and energy decay trend are not affected by the vacancy parameter.
Citation: Hyun Geun Lee. An efficient and energy-stable scheme for the modified phase-field crystal equation with a strong nonlinear vacancy potential[J]. AIMS Mathematics, 2025, 10(3): 5568-5582. doi: 10.3934/math.2025257
In this paper, we present an efficient and energy-stable scheme based on the Crank–Nicolson formula for the modified phase-field crystal equation with a strong nonlinear vacancy potential. In the scheme, the nonlinear terms (the first derivatives of the double-well and vacancy potentials) are treated explicitly, which makes the scheme efficient, and the energy stability is guaranteed by assuming that the second derivatives of the double-well and vacancy potentials are each bounded and by adding two second-order stabilization terms. In particular, by bounding the second derivatives of the double-well and vacancy potentials, respectively, we can choose the stabilization parameters independently of the vacancy parameter. As a result, the convergence constant and energy decay trend are not affected by the vacancy parameter.
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Hyun Geun Lee. An efficient and energy-stable scheme for the modified phase-field crystal equation with a strong nonlinear vacancy potential[J]. AIMS Mathematics, 2025, 10(3): 5568-5582. doi: 10.3934/math.2025257
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