Shock Capturing for High-Order Computational Fluid Dynamics

This thesis contributes to shock-capturing and high-order computational fluid dynamics methods. We are concerned with convection-dominated equations, working towards Navier-Stokes solvers capable of handling large solution gradients. The two high-order methods that we focus on are the novel Active Flux method and the discontinuous Galerkin (DG) method with polynomial order 2 solution reconstruction, which has been gaining popularity in recent years. We present a limiter for the Active Flux method based on the Flux-corrected transport approach. The limiter is designed for acoustics where no maximum principle can be determined, and the solution can become unbounded due to wave focusing. It is capable of reducing and sometimes eliminating overshoots for linear and nonlinear systems. We also suggest ways for an efficient implementation of the evolution step of Active Flux. The Active Flux method produces better quality discontinuous solutions when compared to the DG method while maintaining similar efficiency. The DG method, on the other hand, is more mature and suitable for challenging cases containing shocks, in particular for hypersonic flow. We showcase DG and hybrid DG (HDG) results from transonic to hypersonic flows using a simplified artificial viscosity shock-capturing approach coupled with output-based mesh refinement. Our discoveries include: Mesh Optimization through Error Sampling and Synthesis (MOESS) does not always generate satisfactory results in the presence of strong shocks due to error indicator disparity; an indicator based on the domain integrated total enthalpy balances the refinement on and behind the shock in many cases; HDG has the potential to reduce the cost of solution without compromising the solution quality.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/176458/1/yifanb_1.pd