守恒型Allen-Cahn方程的显式高阶保极值格式

Explicit High-order Maximum Principle Preserving Schemes for the Conservative Allen-Cahn Equation

相比于经典Allen-Cahn方程,修正的Allen-Cahn方程由于加入了非局部的拉格朗日乘子,使得方程解的质量得以守恒.本文针对守恒型Allen-Cahn方程构造一系列最高到八阶精度的保极值格式.基于二阶有限差分空间离散,我们提出一种高阶积分因子两步Runge-Kutta方法求解守恒型Allen-Cahn方程.之后证明该格式可以保持守恒型Allen-Cahn方程的极值原理和质量守恒律,并且给出数值格式的收敛性分析.最后,分别使用二维和三维的数值实验来验证理论结果和数值格式的性能表现.

Compared with the well-known classical Allen-Cahn equation,the modified Allen-Cahn equation,equipped with a nonlocal Lagrange multiplier,enforces the mass conservation for modeling phase transitions.In this paper,a class of up to eighth-order maximum principle preserving schemes are proposed for solving the modified conservative Allen-Cahn equation.Based on the second-order finite-difference space discretization,we investigate the high-order integrating factor two-step Runge-Kutta maximum principle preserving schemes.We prove that the schemes can preserve the maximum principle and mass of the conservative Allen-Cahn equation and give the convergence analysis of proposed schemes.Finally,two-and three-dimensional numerical tests are carried out to verify the theoretical results and demonstrate the performance of proposed schemes.