An Augmented Approach for Stokes Equations With Discontinuous Viscosity and Singular Forces

Abstract

For Stokes equations with discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, the pressure is known to be discontinuous and the velocity is known to be non-smooth. It has been shown that these discontinuities are coupled together which makes it difficult to obtain accurate numerical solutions. In this paper, a second order accurate numerical method that decouples the jump conditions of the fluid variables through two augmented variables has been developed. The GMRES iterative method is used to solve the Schur complement system for the augmented variables which are only defined on the interface. The augmented approach also rescales the Stokes equations in such a way that fast Poisson solvers can be used in each iteration. Numerical examples against exact solutions show that the new method has average second order accuracy in the infinity norm, and the number of GMRES iterations is independent of mesh sizes. An example of a moving interface problem is also presented.

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

“…are explicitly available, the above required Cartesian jump conditions can be systematically derived [12,14], and in Eq. (1.13), the jump contribution c is explicitly known and independent of p. The augmented immersed interface method [3,7] can be used to overcome this difficulty, in which the desirable but unavailable jump conditions are iteratively solved together with the governing equations by GMRES to satisfy the available jump conditions in undesirable forms. In this paper, we propose a different iteration strategy.…”

The immersed interface method for simulating prescribed motion of rigid objects in an incompressible viscous flow

“…Calhoun presented in detail how to set up and solve a linear system for the vorticity sources along a stationary boundary, and proposed to use GMRES to handle a moving boundary. Li et al [21] has implemented a similar idea to impose Neumann pressure boundary conditions in Stokes flows. Le et al [15] employed the same idea in their immersed interface method to enforce the prescribed velocity.…”

From Iim to Augmented Iim: A Powerful Tool for Complex Problems Using Cartesian Meshes

“…The second one (more important one) is that, for some interface problems, an augmented approach may be the only way to derive an accurate algorithm. The augmented techniques enable us to decouple the jump conditions so that the idea of the immersed interface method can be applied, see for example, (Li, 2015;Li, Ito, & Lai, 2007;Li & Qin, 2017).…”