Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 230, Issue 20, Pages 7567-7597Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2011.06.014
Keywords
Poisson equation; Interface jump condition; Ghost Fluid Method; Gradient augmented level set method; High accuracy; Hermite cubic spline
Funding
- National Science Foundation [DMS-0813648]
- Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES - Brazil)
- Fulbright Commission [BEX 2784/06-8]
- NSERC
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0813648] Funding Source: National Science Foundation
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In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard black box solvers, without compromising accuracy. The basic idea of the new approach is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the light-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the standard approaches used to compute the GFM correction terms. In this paper we generalize the GFM correction terms to a correction function, defined on a band around the interface. This function is then shown to be characterized as the solution to a PDE, with appropriate boundary conditions. This POE can, in principle, be solved to any desired order of accuracy. As an example, we apply this new method to devise a 4th order accurate scheme for the constant coefficients Poisson equation with discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil discretization of the Poisson equation, (ii) a representation of the correction function in terms of bicubics, and (iii) a solution of the correction function POE by a least squares minimization. Several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries, its capability to capture sharp discontinuities, and its high convergence rate. (C) 2011 Elsevier Inc. All rights reserved.
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