A Second-Order Relaxation Flux Solver for Compressible Navier-Stokes Equations based on Generalized Riemann Problem Method

TL;DR

This paper introduces a second-order flux solver for compressible Navier-Stokes equations using a hyperbolic relaxation model and generalized Riemann problem method, achieving high accuracy and efficiency across diverse flow regimes.

Contribution

It develops a novel second-order flux solver based on a hyperbolic relaxation model and GRP method, incorporating source terms implicitly for improved stability and adaptability.

Findings

Achieves second-order accuracy in a single stage.

Handles stiff source terms implicitly with linear system solutions.

Demonstrates high resolution across a wide range of flow conditions.

Abstract

In the finite volume framework, a Lax-Wendrof type second-order flux solver for the compressible Navier-Stokes equations is proposed by utilizing a hyperbolic relaxation model. The flux solver is developed by applying the generalized Riemann problem (GRP) method to the relaxation model that approximates the compressible Navier-Stokes equations. The GRP-based flux solver includes the effects of source terms in numerical fluxes and treats the stiff source terms implicitly, allowing a CFL condition conventionally used for the Euler equations. The trade-off is to solve linear systems of algebraic equations. The resulting numerical scheme achieves second-order accuracy within a single stage, and the linear systems are solved only once in a time step. The parameters to establish the relaxation model are allowed to be locally determined at each cell interface, improving the adaptability to diverse flow regions. Numerical tests with a wide range of flow problems, from nearly incompressible to supersonic flows with strong shocks, for both inviscid and viscous problems, demonstrate the high resolution of the current second-order scheme.