We consider solving a generalized Allen-Cahn equation coupled with a passive convection for a given incompressible velocity field. The numerical scheme consists of the first order accurate stabilized implicit explicit time discretization and a fourth order accurate finite difference scheme, which is obtained from the finite difference formulation of the $Q^2$ spectral element method. We prove that the discrete maximum principle holds under suitable mesh size and time step constraints. The same result also applies to construct a bound-preserving scheme for any passive convection with an incompressible velocity field.

翻译：我们考虑解决一个普遍的艾伦-卡恩方程式,同时对特定不可压缩速度场进行被动对流。数字法包括第一级准确稳定、隐含明确时间分解和第四级准确有限差异法,这是从$$2$的光谱元件法的有限差异公式中获得的。我们证明离散最大原则在适当的网格大小和时间步骤限制下存在。同样的结果也适用于为任何带有不可压缩速度场的被动对流构建一个约束性保留方案。