High-Order and Wavelet-Adaptive Immersed Methods for PDEs on Complex Domain Geometries

Abstract

The development of immersed methods brings a promising solution to the numerical simulation of interface-coupled multi-physics problems, such as multi-phase flows and fluidstructure interactions. This renders necessitates the design of novel high-order and efficient solvers based on immersed methods. This thesis examines two pivotal aspects of these methods: firstly, the acceleration of computational processes via adaptive resolution strategies; and secondly, the enhancement of accuracy order while sustaining numerical stability. To achieve the former, we develop a novel wavelet transform algorithm applicable to computational domains with arbitrary geometries. This wavelet transform maintains the order of the wavelet and serves as an indicator for local truncation error (LTE), resulting in an adaptive resolution strategy with explicit error control. To address the latter, we introduce a fifth-order upwind finite difference (FD) scheme that sustains numerical stability across any immersed interface discretization.