Active Flux Methods with Gradient Degrees of Freedom

Abstract

This thesis presents a numerical approach designed to solve wave propagation problems with high accuracy on an unstructured grid. The method employs a fully explicit approach with a compact stencil, addressing both material and remote boundaries automatically. The ultimate goal is to contribute to the development of a highly accurate Navier-Stokes code for aeroacoustic predictions, specifically focusing on waves with wavelengths not significantly larger than the grid size. In essence, the aim is to accurately capture short waves using relatively coarse and cost-effective unstructured grids that can represent intricate geometric details.

The underlying hypothesis is based on the belief that, for ensuring accuracy, particularly in terms of bandwidth – which is often more crucial than achieving high formal accuracy – the method must be fully-discrete and explicit. This choice is driven by the recognition that a numerical stencil, while encompassing the analytical domain of dependence to prevent instability, should not be excessively large. An overly large stencil includes irrelevant data, leading to either instability or an overly dissipative scheme. Moreover, semi-discrete methods, despite enabling high formal accuracy for long waves, prove less effective in ensuring well-behaved mid- and high-frequency behavior due to the gradual inclusion of data from outside the true domain of dependence unless the time step is significantly reduced.

Any endeavor to enhance accuracy invariably involves the processing of additional information, commonly achieved by enlarging the stencil. However, in the scenario of element enlargement, where irrelevant information is incorporated into a large stencil, it not only encounters the issues previously outlined but also poses challenges on irregular grids and in proximity to boundaries. An alternative approach is to augment the set of information retained at each location, as demonstrated in methods such as Discontinuous Galerkin and Active Flux. This extended set of information encompasses various variables, including but not limited to values, gradients, and element averages of the solution.

Guided by these heuristics, the thesis establishes a general framework applicable across arbitrary dimensions and levels of element accuracy. It introduces an analytical solution designed to smooth initial-value wave propagation problems. Consequently, a Hermitian version of the Active Flux method is devised in two dimensions, successfully retaining the advantages of optimal one-dimensional methods. There is a high likelihood that extending this method to three dimensions will be straightforward. Notably, the algorithm autonomously identifies the correct direction of propagating waves, forming a robust foundation for non-reflecting boundary conditions and addressing multi-material problems. The efficacy of the method is demonstrated through various interface conditions in acoustics and elastodynamics wave propagation.

This method attains at least fifth-order accuracy and provides compelling numerical evidence for a form of superconvergence, indicating the potential for highly precise solutions. In doing so, it significantly propels the state-of-the-art in solving wave propagation problems.