Li Wang
Department of Mechanical Engineering
University of Wyoming

Techniques for High-order Adaptive Discontinuous Galerkin Discretizations in Fluid Dynamics

The use of high-order discontinuous Galerkin (DG) discretizations has become more widespread over the last decade for solving convection-dominated computational fluid dynamics problems. The appeal of these methods relates to their favorable asymptotic accuracy properties, combined with compact stencils and favorable scalability properties on parallel computing architectures. This work covers advances in several areas of high-order DG discretizations, including the development of implicit solvers, discrete adjoint methods for output-based error estimation and mesh and time-step adaptation.

For time-dependent problems, high-order implicit time-integration schemes are considered exclusively to avoid the stability restrictions of explicit methods, with particular emphasis on balancing spatial and temporal accuracy of the overall approach. In order to make the high-order schemes competitive, efficient solution techniques consisting of a p-multigrid approach driven by element Jacobi smoothers are investigated and developed to accelerate convergence of the non-linear systems, in which the results demonstrate h independent convergence rates, while remaining relatively insensitive to time-step sizes.

A framework based on discrete adjoint sensitivity analysis has also been developed for applications in shape optimization and goal-oriented error estimation. An adaptive discontinuous Galerkin algorithm driven by an adjoint-based error estimation procedure has been developed, which incorporates both h-, p- and combined hp-adaptive schemes, for producing accurate simulations at optimal cost in the objective functional of interest. Current results show superior performance of these adaptive schemes over uniform mesh refinement methods, as well as the potential of the hp refinement approach to capture strong shocks without limiters. Finally, the adjoint-based error estimation strategy is successfully extended to unsteady flow problems, where the time-dependent flow solution is solved in a forward manner in time but the corresponding unsteady adjoint solution is evaluated as a backward time integration. Results demonstrate that this methodology provides accurate global temporal error prediction, and may be employed to drive an adaptive time-step refinement strategy for improving the accuracy of specified time-dependent functionals of interest.