On the relative merits of interpolation schemes for the immersed boundary method: a case study with the heat equation

Abstract

In this work, three implementation options for the ghost cell immersed boundary method are compared. These options are alternatives to a rather common implementation of the method that is susceptible to numerical instability in the calculation of the bilinear interpolation in some cases. The method is implemented for a second-order spatial discretization of the heat equation in a non-rectangular domain and the errors for each option are analyzed in terms of the order of accuracy and the way they are distributed in the domain. The best option, which was the only one to maintain the second order of convergence of the discretization, is to consider non-symmetric extrapolation with bilinear interpolation, instead of using inverse distance weighted interpolation with symmetric or non-symmetric extrapolation.