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<p>This paper proposes a new functional associated with the time-fractional Allen-Cahn equation. The functional is dissipative and provides a new perspective to explore the dissipative property of the classical energy functional. We assert that the proposed functional is consistent as the differential order approaches 1, and that it exhibits asymptotic behavior with the trends of the classical functional. Using <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( L_1 \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> method integrating with different convex splitting schemes to deal with the nonlinear source term, we establish that the numerical scheme maintains the dissipation property and maximum principle unconditionally. Our convergence analysis confirms that linear and nonlinear convex splitting schemes achieve similar accuracy, with both methods displaying order consistency with the fractional differential operator. This demonstrates the effectiveness of linear convex splitting methods in solving the time-fractional Allen-Cahn equation.</p>

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An Auxiliary Dissipative Functional and Numerical Analysis with \(L_1\) methods and Convex Splitting Method to Time Fractional Allen-Cahn Equation

  • 期刊论文

  • 发表日期：

    2025年11月3日

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  • Bing-Ze Lu,
  • Yu-Chen Shu

摘要

This paper proposes a new functional associated with the time-fractional Allen-Cahn equation. The functional is dissipative and provides a new perspective to explore the dissipative property of the classical energy functional. We assert that the proposed functional is consistent as the differential order approaches 1, and that it exhibits asymptotic behavior with the trends of the classical functional. Using \( L_1 \) L 1 method integrating with different convex splitting schemes to deal with the nonlinear source term, we establish that the numerical scheme maintains the dissipation property and maximum principle unconditionally. Our convergence analysis confirms that linear and nonlinear convex splitting schemes achieve similar accuracy, with both methods displaying order consistency with the fractional differential operator. This demonstrates the effectiveness of linear convex splitting methods in solving the time-fractional Allen-Cahn equation.