Implicit gradients based conservative numerical scheme for compressible flows

Abstract

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cellcentered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new approach has superior dispersion and dissipation properties in comparison to the compact reconstruction approach. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method's capability using the implicit gradient method for compressible flows.

Gradient Based Reconstruction: Inviscid and viscous flux discretizations, shock capturing, and its application to single and multicomponent flows

“…The exact solution is computed by an exact Riemann solver [71] and the specific heat ratio for this test case is 5 3 . Numerical results for density and velocity obtained for MEG6, MIG4 and TENO5 schemes on a N = 900 grid for the final time t = 6 are shown in Fig.…”

“…On the other hand, in the implicit gradient schemes previously developed in Ref. [5], the sharing is possible and done because we employ solution reconstructions, expressed in terms of gradients and Hessians, for both inviscid and viscous discretizations. Such a construction is common in unstructured-grid discretizations, but often the solution gradients need to be computed by two different methods for nonlinear-solver stability (unweighted least-squares method for inviscid fluxes) [6].…”