Active Flux Schemes.

Abstract

This dissertation details the development of active flux schemes, a new class of methods for solving conservation laws. Active flux methods address three issues plaguing production-level computational fluid dynamics (CFD) codes: reliance on one-dimensional Riemann solvers, second-order accuracy, and computational stencils that do not easily parallelize. The key concept is that edge and vertex values are updated and evolved independently from the conserved cell-average quantities. Interface values are then used to calculate fluxes that conservatively update the cell-averages. Because the edge updates do not need to be conservative, any convenient method can be used and proper attention can be given to multidimensional physics. The scheme uses parabolic reconstructions, with a cubic bubble function to maintain conservation in two dimensions, making it third-order accurate by construction. All of the reconstructions and updates are local to a single element, giving AF schemes a very compact stencil suitable for parallelization. Additionally, the AF method is fully discrete, advancing from time-level n to n + 1 in a single step. The method is demonstrated on the linear advection, linear acoustics, and linearized Euler equations in one and two dimensions. The AF method has several advantages over more traditional schemes. For one, the extra degrees of freedom within the cell mean that frequencies up to 2π can be resolved, which is double the frequency range for comparable finite volume (FV) schemes. The AF scheme has superior dissipation and dispersion properties, especially as the Courant number approaches one. Its compact stencil makes the AF solution far less sensitive to irregular meshes than a third-order FV scheme. The AF scheme economically achieves third-order accuracy using two degree(s) of freedom (DOF) per element in one dimension and three DOF in two dimensions. This is comparable to the DOF in a discontinuous Galerkin scheme using linear reconstructions. The AF method achieves third-order accuracy for all of the equation sets using randomized, unstructured meshes. The multidimensional treatment of the acoustics system allows the AF method to preserve excellent symmetry properties on an irregular triangular mesh.