Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces
Xiaohai Wan Zhilin Li
Discrete & Continuous Dynamical Systems - B 2012, 17(4): 1155-1174 doi: 10.3934/dcdsb.2012.17.1155
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
Thomas P. Witelski David M. Ambrose Andrea Bertozzi Anita T. Layton Zhilin Li Michael L. Minion
Discrete & Continuous Dynamical Systems - B 2012, 17(4): i-ii doi: 10.3934/dcdsb.2012.17.4i
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.

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An augmented immersed interface method for moving structures with mass
Jian Hao Zhilin Li Sharon R. Lubkin
Discrete & Continuous Dynamical Systems - B 2012, 17(4): 1175-1184 doi: 10.3934/dcdsb.2012.17.1175
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.
keywords: Immersed interface method Navier-Stokes moving interface implicit scheme augmented method fluid-structure. projection method

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