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Solving immersed interface problems using a new interfacial points-based finite difference approach. (English) Zbl 1448.65195

Summary: A new finite difference scheme based on the higher-order compact technique is presented for solving problems with complex immersed interfaces in arbitrary dimensions. The scheme is designed for general interface problems in which the coefficients, the source term, the solution, and its normal flux may be discontinuous across the interface. The originality of the scheme lies in the use of additional values at interfacial points (points at which grid lines intersect the interface) as nodes in the stencil, which allows straightforward use of the standard finite difference approximations. Appropriate interpolation techniques are used on both sides of the interface to determine the interfacial values. Numerical tests are carried out to validate the scheme for solving elliptic equations in both two and three dimensions, which show that the proposed scheme has overall second-order accuracy. The scheme thus developed is also applied to solve incompressible, two-dimensional Stokes flows. In this process, we compare our computed results with the results obtained from existing schemes, and in most of the cases, our scheme is found to produce relatively better results.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
52B10 Three-dimensional polytopes
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
68U07 Computer science aspects of computer-aided design
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
65D05 Numerical interpolation
76M20 Finite difference methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
35Q35 PDEs in connection with fluid mechanics
35R05 PDEs with low regular coefficients and/or low regular data
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