A high-order 3D immersed interface finite difference method for the advection-diffusion equation

Abstract

We present a finite-difference based immersed interface method for the high-order discretization of 3D advection-diffusion problems on regular Cartesian grids. Our approach efficiently handles convex and non-convex geometries using a weighted least squares polynomial reconstruction algorithm. We analyze the stability of the approach for 2D and 3D parabolic and hyperbolic problems and demonstrate stable convergence results at third-order for advection and at fourth and sixth order for diffusion problems. Our immersed interface approach naturally handles one-sided Dirichlet or Neumann boundary conditions as well as two-sided jump boundary conditions within the same framework, opening the door to high-order treatment of 3D interface-coupled multiphysics problems. We demonstrate the capability of our approach using a 3D conjugate heat transfer problem resolved with third-order accuracy on a multi-resolution adaptive grid.