Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size.

中文翻译：

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不规则域或接口上亥姆霍兹方程的一些新的有限差分方法。

有效地求解亥姆霍兹方程 Δu + λu = f 是许多应用的挑战。例如，许多不可压缩 Navier-Stokes 方程的高效求解器的核心部分是求解一个或多个 Helmholtz 方程。在本文中，提出了两种新的有限差分方法来求解不规则域或界面上的亥姆霍兹方程。对于不规则域上的亥姆霍兹方程，当 λ 的大小较大时，使用现有的增强浸入界面方法 (AIIM) 获得的数值解的精度可能会下降。在我们的新方法中，在应用 AIIM 之前，我们使用水平集函数将源项和 PDE 扩展到更大的域。针对具有界面的亥姆霍兹方程，提出了一种新的最大原理保持有限差分法。新方法仍然使用标准的五点模板，并在不规则网格点处修改了有限差分方案。得到的有限差分方程线性系统的系数矩阵满足离散最大值原理的符号属性，并且可以使用多重网格求解器有效地求解。有限差分方法还扩展到处理时间离散方程，其中解系数 λ 与网格大小成反比。