A High-Order Immersed Interface Method for 3D Transport Equations

Immersed Interface Methods (IIMs) are a class of Cartesian grid-based finite difference schemes that can achieve high order accuracy while allowing for non-conforming domain boundaries or material interfaces. These schemes are particularly promising for simulations of interface-coupled multiphysics problems. However, many existing IIMs have focused on second-order methods and simple 2D geometries, and their extension to higher-order methods and complex 3D geometries remains a challenge. In this presentation we demonstrate a high-order immersed interface method for 3D simulations with smooth non-convex domain boundaries or material interfaces. The method allows for third order spatial accuracy for advection terms and up to sixth order spatial accuracy for diffusion terms. This IIM discretization is implemented in MURPHY, an open-source software framework for adaptive mesh refinement, which allows for massively parallel simulations of PDEs with complex 3D interfaces. We confirm the stability and accuracy of our IIM through simulations of 3D advection-diffusion with a variety of boundary conditions on irregular domains and material interfaces, and showcase the ability of the method to handle more complex nonlinear PDEs with nontrivial dynamics on both sides of a material interface.