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<p>We consider the numerical approximations of the ternary Allen-Cahn equations with either the Ginzburg-Landau polynomial potential or Flory-Huggins logarithmic potential. It is particularly challenging to design accurate temporal discretizations that preserve energy stability and the maximum bound principle (MBP) while achieving first- or second-order temporal accuracy. To address this, we develop a set of first- and second-order numerical schemes based on exponential time differencing Runge-Kutta (ETDRK) methods. We prove that the proposed schemes unconditionally preserve the initial energy decreasing property and MBP for the ternary Allen-Cahn equations. Based on the MBP, we further establish a maximum-norm error estimate. To the best of our knowledge, this is the first work to show that ETDRK schemes can unconditionally guarantee MBP and the discrete energy stability for a ternary Allen-Cahn model. Finally, various numerical experiments are performed to verify the proposed schemes.</p>

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Some maximum bound principle preserving and energy stable exponential time differencing Runge-Kutta schemes for ternary Allen-Cahn equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月10日

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  • Xinyang Wu,
  • Hongbo Chen,
  • Zian Wang,
  • Tianliang Hou

摘要

We consider the numerical approximations of the ternary Allen-Cahn equations with either the Ginzburg-Landau polynomial potential or Flory-Huggins logarithmic potential. It is particularly challenging to design accurate temporal discretizations that preserve energy stability and the maximum bound principle (MBP) while achieving first- or second-order temporal accuracy. To address this, we develop a set of first- and second-order numerical schemes based on exponential time differencing Runge-Kutta (ETDRK) methods. We prove that the proposed schemes unconditionally preserve the initial energy decreasing property and MBP for the ternary Allen-Cahn equations. Based on the MBP, we further establish a maximum-norm error estimate. To the best of our knowledge, this is the first work to show that ETDRK schemes can unconditionally guarantee MBP and the discrete energy stability for a ternary Allen-Cahn model. Finally, various numerical experiments are performed to verify the proposed schemes.