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DCDS-B

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.

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