Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces
Xiaohai Wan Zhilin Li
Solving a Helmholtz equation $\Delta u + \lambda u = f$ 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 $\lambda$ 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 $\lambda$ is inversely proportional to the mesh size.
keywords: augmented IIM. immersed interface method Helmholtz equation irregular domain discrete maximum principle elliptic interface problem finite difference method level set function embedding method

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