For reaction-diffusion equations in irregular domain with moving boundaries, the numerical stability constraints from the reaction and diffusion terms often require very restricted time step size, while complex geometries may lead to difficulties in accuracy when discretizing the high-order derivatives on grid points near the boundary. It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties. Applying an implicit scheme may be able to remove the stability constraints on the time step, however, it usually requires solving a large global system of nonlinear equations for each time step, and the computational cost could be significant. Integration factor (IF) or exponential differencing time (ETD) methods are one of the popular methods for temporal partial differential equations (PDEs) among many other methods. In our paper, we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries. In particular, we rewrite all ETD schemes into a linear combination of specific {\phi}-functions and apply one start-of-the-art algorithm to compute the matrix-vector multiplications, which offers significant computational advantages with adaptive Krylov subspaces. In addition, we extend this method by incorporating the level set method to solve the free boundary problem. The accuracy, stability, and efficiency of the developed method are demonstrated by numerical examples.
翻译:对于在有移动边界的非常规领域反扩散方程式而言,反应和传播术语中的数字稳定性限制往往要求非常有限的时间步数,而复杂的地理比例在边界附近的网格点上将高阶衍生物分解时,可能会导致精确性方面的困难。在设计数字方法时,很难有效和准确地处理两种困难。应用一个隐含的方案可能能够消除时间步骤上的稳定性限制,然而,通常需要解决一个大型的全球非线性方程式系统,每个时间步骤的非线性方程式,计算成本可能很高。整合系数(IF)或指数性差异时间(ETD)法是时间部分偏差方程式(PDEs)许多其他方法中流行的方法之一。在我们的文件里,我们把ETD方法与嵌入式边界法结合起来,用复杂的地理模型解决反应-扩散方程式系统。特别是,我们将所有ETD方法改写成一个特定时间步骤的线性组合,并应用一种开始式的算法来计算矩阵变量多变法(ETD),这是用于时间偏差方方方方方程的流行方法的一种方法。我们用自由的计算方法将稳定性方法的精确性方法纳入了我们确定的数字方法。