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A direct IIM approach for two-phase Stokes equations with discontinuous viscosity on staggered grids. (English) Zbl 1410.76281

Summary: In this paper, a direct immersed interface method (IIM) is proposed to solve two-phase incompressible Stokes equations with an interface and a piecewise constant viscosity on staggered grids. The velocity components and the pressure are placed in different grid points and the marker and cell (MAC) scheme is used for discretizing the momentum and continuity equations at regular grid points. At irregular grid points, correction terms are added to the finite difference scheme to offset the discontinuities. The correction terms are determined directly by an interpolation scheme using the values of both the velocity and pressure at nearby grid points. The resulted linear system of equations is rank-one deficient and is solved by the Uzawa iterative method. In each Uzawa iteration, an inner GMRES solver is used and preconditioned by the discrete Laplacian. The computed numerical solutions are second order accurate in the \(L_\infty\) norm for both the velocity and pressure, which is demonstrated in numerical tests. Compared with the augmented interface method (AIIM), one of advantages of this approach is that it avoids the costs for introducing augmented variables and difficulties in solving them from the corresponding Schur complement system. Hence, this new method is easier to implement and computationally more efficient.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65N06 Finite difference methods for boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
35R05 PDEs with low regular coefficients and/or low regular data
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows

Software:

IIMPACK
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Full Text: DOI

References:

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