A novel concept for the design of immersed interface methods

Abstract

The objective of this paper is to present a novel, robust, high-order accurate Immersed Interface Method (IIM) for advection–diffusion type equations. In contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability, we combine local Taylor-series expansion at irregular grid points with a local stability constraint as part of the design process. Stability investigations of the IIM are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method, as long as the Neumann number is less than its limiting value. One of the key aspects of this IIM is that the irregular finite-difference stencils can be isolated from the rest of the computational domain. To validate our novel immersed interface approach, two-dimensional and three-dimensional test cases for model equations are presented. In addition, this method is applied to the incompressible Navier–Stokes equations to conduct stability investigations of a boundary layer flow over a rough surface, and for investigations of pulsatile stenotic flows. Stability investigations of wall bounded flows are challenging for immersed methods, because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms, in particular regarding the wave relation between the wave velocity components close to the wall.