论文标题
能量加上最大绑定的lunge-kutta方法
Energy plus maximum bound preserving Runge-Kutta methods for the Allen-Cahn equation
论文作者
论文摘要
很难设计高阶数值方案，这些方案可以保留某些相位场方程的最大绑定特性（MBP）和能量耗散定律。在过去的几十年中，已经开发了强稳定性（SSP）runge-kutta方法用于双曲偏微分方程的数值解，在这种情况下，强稳定性意味着不侵入基础解决方案的最大结合。但是，现有的SSP RK方法框架无法处理诸如耗能法律之类的非线性稳定性。这项工作的目的是扩展该SSP理论，以处理艾伦-CAHN类型的非线性相位场方程，该类型通常满足最大结合保留（MBP）和能量耗散定律。更确切地说，对于Runge-Kutta时间离散，我们首先得出了满足MBP的一般必要条件；我们进一步提供了必要的条件，在该条件下，MBP方案满足了能量耗散。
It is difficult to design high order numerical schemes which could preserve both the maximum bound property (MBP) and energy dissipation law for certain phase field equations. Strong stability preserving (SSP) Runge-Kutta methods have been developed for numerical solution of hyperbolic partial differential equations in the past few decades, where strong stability means the non-increasing of the maximum bound of the underlying solutions. However, existing framework of SSP RK methods can not handle nonlinear stabilities like energy dissipation law. The aim of this work is to extend this SSP theory to deal with the nonlinear phase field equation of the Allen-Cahn type which typically satisfies both maximum bound preserving (MBP) and energy dissipation law. More precisely, for Runge-Kutta time discretizations, we first derive a general necessary and sufficient condition under which MBP is satisfied; and we further provide a necessary condition under which the MBP scheme satisfies energy dissipation.
