A Residual-Based Procedure for hp-Adaptation on 2D Hybrid Meshes

H- and p-adaptations are effective ways of achieving high-quality solutions for CFD simulations, with reasonable computing costs. Combined hp-adptation offers even greater flexibility for complex problems. In the present study, a residual based procedure for hpadaptaion is developed. The adaptation is driven by an efficient multi-p a posteriori error estimator. P-adaptation is applied to smooth regions of the flow field while h-adaptation targets the non-smooth regions, identified by accuracy-preserving TVD marker. Several numerical tests with the correction procedure via reconstruction (CPR) formulation are presented to demonstrate the capability of the technique. I. Introduction daptive solution methods have been an active topic of CFD research for decades. Through these methods, computing power can be redistributed to where it is needed the most, thus the desired accuracy can be achieved with minimal costs. For a high-order method, two types of adaptations are usually considered: local mesh subdivision (h-refinement) and local polynomial degree enrichment (p-refinement). It has been shown that p-adaptation is superior in regions where the solution is smooth [13], while h-adaptation is preferred in non-smooth regions such as flow discontinuities, where high-order methods become less effective. Obviously, the combination of both h- and padaptation offers greater flexibility than methods that incorporates h- or p-adaptation alone. Hp-adaptation is made possible by the development of a series of compact high-order methods for unstructured meshes, such as the discontinuous Galerkin (DG) method [3,4,7], the spectral volume (SV) method [14,21] and the spectral difference (SD) method [15]. Unlike the finite volume method that achieve high-order by expanding the stencil, the above methods employ a polynomial to approximate the solution in each cell or element, and the polynomials collectively form a function which is discontinuous across cell boundaries. The compact solution reconstruction of these methods is an ideal feature for p-adaptation. Recently, the correction procedure via reconstruction (CPR) formulation was developed in 1D [11], and extended to simplex amd hybrid meshes in [22]. It unites the DG, SV and SD method into a nodal differential formulation. The CPR formulation combines the compactness and high accuracy with the simplicity and efficiency of the finite difference method, and can be easily implemented for arbitrary unstructured meshes. The current hp-adaptation is based on the CPR formulation, but similar methods can be easily applied to other campact high-order methods.