High Order Immersed Finite Difference Methods for Complex Domains with Moving Boundaries and Interfaces

Abstract

Moving domain boundaries and material interfaces are a hallmark of multiphysics systems such as fluid-structure interaction, alloy solidification, and multiphase flows. Simulating moving interfaces with traditional techniques requires a moving mesh that continuously adapts to the interface, which is costly and places restrictions on the interface motion. Immersed methods avoid these challenges by simulating moving geometries on a stationary Cartesian grid, locally altering the numerical method to account for boundaries and interfaces that are not grid-aligned. Most existing immersed methods have low-order spatial accuracy, requiring fine grids to generate accurate results. High order immersed methods can produce more accurate results at lower resolution, making them a promising tool for 3D simulations with tight error tolerances. However, the majority of available high order immersed methods have been numerical experiments developed for stationary 2D geometries and simple PDEs. In this thesis we demonstrate that high order immersed methods can be extended to complex nonlinear PDEs and moving 3D geometries, both of which are necessary to simulate practical engineering problems. We begin by introducing a boundary treatment that locally approximates PDE solutions with high order accuracy using a weighted least-squares fit, and show that the procedure remains valid for smooth 2D or 3D geometries satisfying a local curvature constraint. This boundary treatment is combined with a high order finite difference method to discretize the Poisson equation with up to sixth order accuracy. We then expand the scope of the method to include PDEs with immersed material interfaces, spatially-variable coefficients, vector-valued unknowns, cross-derivative terms, and nonlinearities. These techniques are applied to generate a sixth-order discretization of 2D nonlinear elasticity, demonstrating the applicability of high order immersed methods to complex PDE systems relevant in mechanical engineering. In the second half, we focus on large-scale 3D simulations with moving boundaries. We construct a third order immersed advection discretization with provable stability in one dimension, and show experimentally that the scheme remains stable in 2D and 3D domains. To treat moving boundaries, we introduce a general framework that allows high order immersed methods to maintain their accuracy in both space and time when paired with any explicit Runge-Kutta time integrator. We conclude by presenting results from massively-parallel high order simulations of the 3D advection-diffusion equation with moving boundaries on a multiresolution grid. Taken together, these results demonstrate that high order immersed methods can achieve the scale and complexity necessary to enable practical simulations that are difficult or impossible with traditional mesh-based techniques.