Implicit gradients based conservative numerical scheme for compressible flows

TL;DR

This paper presents a new conservative numerical scheme for compressible flows that uses high-order implicit gradients computed by compact finite differences and incorporates a shock capturing method to handle discontinuities effectively.

Contribution

It introduces a novel implicit gradient approach combined with a BVD shock capturing algorithm for improved accuracy in compressible flow simulations.

Findings

Accurate computation of fluxes using implicit gradients.

Effective suppression of oscillations near shocks.

Validated performance on multiple test cases.

Abstract

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered 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. 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.