Comparison of h- and p-Derived Output-Based Error Estimates for Directing Anisotropic Adaptive Mesh Refinement in Three-Dimensional Inviscid Flows

Abstract

The use of adjoint-based error estimation in conjunction with a highly parallel and scalable, anisotropic, block-based, adaptive mesh refinement (AMR) technique is considered for the more efficient prediction of three-dimensional compressible flows. In particular, a comparison is made between the computational performances of output-based error estimates, derived via two approaches, namely one based on mesh (or h) refinement and the other based on order (or p) refinement, for directing the mesh refinement in anisotropic AMR scheme. The AMR scheme allows enhancement of local mesh resolution, with preference given to directions as dictated by the flow solution. The proposed adjoint-based error estimation technique provides a posteriori estimates of the error for an engineering functional of interest in terms of estimates of the local solution error following from the solution residual. The estimated error in the solution residual is obtained either via direct refinement of the mesh in the preferred directions (here referred to as the h-derived error indicator) or by using a higher-order spatial operator with anisotropic feature detection based on the anisotropic smoothness indicator of an appropriate solution quantity (here referred to as the p-derived error indicator). Both approaches are considered here. Additionally, two formulations of the adjoint-based error indicator are examined for directing the output-based AMR. The first is the so-called computable correction (CC), where the residual error is weighted by the corresponding adjoint solution for the functional of interest, and the second is the so-called error in the computable correction (ECC), which is comprised of a linear combination of the residual error weighted with the adjoint solution and the adjoint residual weighted with the primal solution. The resulting output error indicator is used to direct the mesh refinement, with regions of the solution domain contributing most significantly to the functional error being selected for local enrichment of the mesh. In this way, the computed accuracy of the functional is increased while potentially greatly reducing the associated computational cost of performing the simulation. For the cases of interest, both low-and high-order upwind finite-volume spatial discretization schemes are applied in conjunction with the block-based AMR scheme to the solution of the partial differential equations governing steady-state inviscid compressible flows. The potential benefits of the proposed anisotropic block-based AMR with adjoint-based error estimation are demonstrated for a range of compressible inviscid flow problems of varying complexity. Comparisons of solution accuracy and relative computational costs for results obtained using both hand p-derived error estimates of the solution residual are examined and discussed.