On the Acoustic Component of Active Flux Schemes for Nonlinear Hyperbolic Conservation Laws

Abstract

Current numerical methods used in production-level CFD codes are found to be lacking in many respects; they are only second-order accurate, rely on inherently one-dimensional solvers, and are ill-equipped to handle more complex fluid flow problems such as turbulence, aeroacoustics and vortical flows just to name a few. Recently, a new class of third-order methods known simply as Active Flux (AF) has been introduced to address some of these issues. The AF method is best understood as a finite-volume method with additional degrees of freedom (DOF) at the interface to independently evolve interface fluxes. It is a fully discrete, maximally stable method that uses continuous data representation, and because the interface fluxes are computed independently from the cell-average values, true multidimensional solvers can be used.

This dissertation focuses on the development of the AF method aimed at solving conservation laws describing acoustic processes. The method is demonstrated for linear and nonlinear acoustic equations in two-dimensions as well as for the full Euler equations where we employ operator splitting between the advective and acoustic processes. Given its continuous representation, the AF method economically achieves third-order accuracy using only three DOF in two dimensions, which is comparable to the discontinuous Galerkin method using linear reconstruction (DG1). A direct comparison between the two methods for acoustic problems finds that the AF method is capable of matching the accuracy of DG1 with a mesh spacing about three times greater and uses time steps about 2.5 times longer. The AF solutions also display superior circular symmetry with significantly less scatter than DG1, which we attribute to the method being able to employ truly multidimensional solvers. In addition, we find that on the same grid and to achieve the same level of error, the computation time for the AF method is more than one magnitude less than DG1 and approximately 3 to 5 times less than DG with quadratic reconstruction (DG2). Finally, various boundary conditions are introduced and developed for the AF scheme including far-field and curved wall boundaries.