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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.

DCDS-B

Studies of problems in fluid dynamics have spurred research in
many areas of mathematics, from rigorous analysis of nonlinear partial
differential equations, to numerical analysis, to modeling
and applied analysis of related physical systems.
This special issue of Discrete and Continuous Dynamical Systems Series B
is dedicated to our friend and colleague Tom Beale in recognition of
his important contributions to these areas.

For more information please click the "Full Text" above.

For more information please click the "Full Text" above.

keywords:

DCDS-B

We present an augmented immersed interface method for simulating
the dynamics of a deformable structure with mass in an incompressible fluid.
The fluid is modeled by the Navier-Stokes equations in two dimensions.
The acceleration of the structure due to mass is coupled with the flow velocity and the pressure.
The surface tension of the structure is assumed to be a constant for simplicity.
In our method, we treat the unknown acceleration as the only augmented variable
so that the augmented immersed interface method can be applied.
We use a modified projection
method that can enforce the pressure jump conditions corresponding to the unknown
acceleration. The acceleration must match the flow acceleration along the interface.
The proposed augmented method is tested against an exact solution with a stationary interface.
It shows that the augmented method has a second order of convergence in space.
The dynamics of a deformable circular structure with mass is also investigated.
It shows that the fluid-structure system has bi-stability: a stationary state for a smaller
Reynolds number and an oscillatory state for a larger Reynolds number.
The observation agrees with those in the literature.

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