An Iterative Two-Fluid Pressure Solver Based on the Immersed Interface Method

An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm. The iteration is constructed by introducing an unsteady term in the pressure Poisson equation. In each iteration step, a Helmholtz equation is solved on the Cartesian grid using FFT. The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across two- fluid interfaces. This solver can also be used to solve Poisson equations on irregular domains. AMS subject classifications: 76T99, 35J05, 65N06

[1]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[2]  H. Fasel,et al.  A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains , 2005 .

[3]  Z. Jane Wang,et al.  Systematic Derivation of Jump Conditions for the Immersed Interface Method in Three-Dimensional Flow Simulation , 2005, SIAM J. Sci. Comput..

[4]  Sheng Xu,et al.  The immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow , 2008, J. Comput. Phys..

[5]  Zhilin Li,et al.  The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics) , 2006 .

[6]  Z. Jane Wang,et al.  An immersed interface method for simulating the interaction of a fluid with moving boundaries , 2006, J. Comput. Phys..

[7]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[8]  Z. J. Wang,et al.  A 3D immersed interface method for fluid–solid interaction , 2008 .

[9]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[10]  D. Juric,et al.  A front-tracking method for the computations of multiphase flow , 2001 .

[11]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[12]  Randall J. LeVeque,et al.  Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension , 1997, SIAM J. Sci. Comput..

[13]  ChorinAlexandre Joel.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[14]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[15]  Zhilin Li,et al.  Convergence analysis of the immersed interface method , 1999 .

[16]  Sheng Xu,et al.  DERIVATION OF PRINCIPAL JUMP CONDITIONS FOR THE IMMERSED INTERFACE METHOD IN TWO-FLUID FLOW SIMULATION , 2009 .

[17]  Kazufumi Ito,et al.  A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains , 2009, J. Comput. Phys..