We describe a method for solving the two-dimensional Navier–Stokes equations in irregular physical domains. Our method is based on an underlying uniform Cartesian grid and second-order finite-difference/finite-volume discretizations of the streamfunction-vorticity equations. Geometry representing stationary solid obstacles in the flow domain is embedded in the Cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells. Along the embedded boundary, we determine a distribution of vorticity sources needed to impose the no-slip flow conditions. This distribution appears as a right-hand-side term in the discretized fluid equations, and so we can use fast solvers to solve the linear systems that arise. To handle the advective terms, we use the high-resolution algorithms in CLAWPACK. We show that our Stokes solver is second-order accurate for steady state solutions and that our full Navier–Stokes solver is between first- and second-order accurate and reproduces results from well-studied benchmark problems in viscous fluid flow. Finally, we demonstrate the robustness of our code on flow in

Abstract: A new numerical method, called the Explicit Simplified Interface Method (ESIM), is developed in the context of acoustic wave propagation in heterogeneous media. Equations of acoustics are written as a first-order linear hyperbolic system. Away from interfaces, a standard scheme (Lax-Wendroff, TVD, WENO...) is used in a classical way. Near interfaces, the same scheme is used, but it is applied on a set of modified values deduced from numerical values and from jump conditions at interfaces. It amounts to modify the scheme so that its order of accuracy is maintained at irregular points, despite the non-smoothness of the solution. This easy to implement interface method requires few additional computational resources and it can be applied to other partial differential equations. Keywords: Acoustics; heterogeneous media; Lax-Wendroff, TVD and WENO schemes; Interface Methods; discontinuous coefficients. AMS Classification: 65M06, 35L05

Abstract. In this paper, numerical methods are proposed for some interface problems in polar or Cartesian coordinates. The new methods are based on a formulation that transforms the interface problem with a non-smooth or discontinuous solution to a problem with a smooth solution. The new formulation leads to a simple second order finite difference scheme for the partial differential equation and a new interpolation scheme for the normal derivative of the solution. In conjunction with the fast immersed interface method, a fast solver has been developed for the interface problems with piecewise constant but discontinuous coefficient using the new formulation in polar coordinate system.