In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unkn...

The paper presents a two-dimensional immersed interface technique for the vortex-in-cell (VIC) method for the simulation of flows past complex geometries. The particle–mesh VIC algorithm is augmented by a local particle– particle (PP) correction term in a particle–particle particle–mesh (PPPM) context to resolve sub-grid scales incurred by the presence of the immersed interface. The PP correcti...

An immersed boundary method for computing viscous, subsonic compressible flows with complex shaped stationary immersed boundaries is presented. The method employs a ghost-cell technique for imposing the boundary conditions on the immersed boundaries. The current approach leads to a sharp representation of the immersed boundaries, a property that is especially useful for flow simulations at high...

We present effective and efficient matrix-free preconditioning techniques for the augmented immersed interface method (AIIM). AIIM has been developed recently and is shown to be very effective for interface problems and problems on irregular domains. GMRES is often used to solve for the augmented variable(s) associated with a Schur complement A in AIIM that is defined along the interface or the...

In immersed interface methods, solids in a fluid are represented by singular forces in the Navier–Stokes equations, and flow jump conditions induced by the singular forces directly enter into numerical schemes. This paper focuses on the implementation of an immersed interface method for simulating fluid–solid interaction in 3D. The method employs the MAC scheme for the spatial discretization, t...

Article history: Received 21 August 2015 Received in revised form 29 February 2016 Accepted 16 April 2016 Available online 19 April 2016

A new simple fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized with a good accuracy on a compact stencil. Auxiliary unknowns are created at existing grid locations to increase the degrees of freedom of the initia...

Article history: Received 28 February 2014 Received in revised form 15 August 2014 Accepted 5 November 2014 Available online 11 November 2014

An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid. 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 s...

This paper introduces a novel high order interface scheme, the matched interface and boundary (MIB) method, for solving elliptic equations with discontinuous coefficients and singular sources on Cartesian grids. By appropriate use of auxiliary line and/or fictitious points, physical jump conditions are enforced at the interface. Unlike other existing interface schemes, the proposed method disas...