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

Tuowei Chen, Zhifang Du

TL;DR

The paper develops a Lax-Wendroff-type, second-order flux solver for the 2-D compressible Navier-Stokes equations by embedding a hyperbolic relaxation formulation into a generalized Riemann problem framework. Stiff viscous terms are handled implicitly through GRP-based time derivatives, enabling a standard Euler-like CFL condition and solving a sparse linear system once per time step. Local interface definitions of the relaxation parameters $a$ and $\epsilon$ enhance robustness across smooth and shocked regions, while avoiding explicit evolution of relaxation variables. Extensive inviscid and viscous test cases demonstrate high-resolution performance, accuracy, and stability, with potential pathways toward higher-order extensions and applications to other convection-diffusion systems.

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.

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

TL;DR

The paper develops a Lax-Wendroff-type, second-order flux solver for the 2-D compressible Navier-Stokes equations by embedding a hyperbolic relaxation formulation into a generalized Riemann problem framework. Stiff viscous terms are handled implicitly through GRP-based time derivatives, enabling a standard Euler-like CFL condition and solving a sparse linear system once per time step. Local interface definitions of the relaxation parameters and enhance robustness across smooth and shocked regions, while avoiding explicit evolution of relaxation variables. Extensive inviscid and viscous test cases demonstrate high-resolution performance, accuracy, and stability, with potential pathways toward higher-order extensions and applications to other convection-diffusion systems.

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.

Paper Structure

This paper contains 26 sections, 78 equations, 12 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Governing equations and relaxation model
  3. Finite volume scheme based on hyperbolic relaxation model
  4. Second-order finite volume scheme
  5. Set-up of the generalized Riemann problem
  6. Generalized Riemann problem solver for the relaxation model
  7. Associated Riemann solver
  8. Generalized Riemann problem solver
  9. Solving the linear system of algebraic equations
  10. Parameters
  11. Summary of the algorithm
  12. Numerical experiments
  13. Inviscid flows
  14. Advection of a density perturbation
  15. Isentropic vortex propagation problem
  16. ...and 11 more sections

Figures (12)

  • Figure 1: The density distribution for the first 2-D Riemann problem at $t=0.2$ (left, 30 uniform contours are drawn) and the section of the density along $y=x$ (right).
  • Figure 2: The density distribution for the second 2-D Riemann problem at $t=0.25$ (left) and the local enlargement (right). 20 uniform contours are drawn.
  • Figure 3: Double Mach reflection problem: the density contours at $t=0.2$ with $960 \times 240$ (top) and $1920\times 480$ (bottom) cells. 30 uniform contours are displayed.
  • Figure 4: Double Mach reflection problem: enlarged density contours around the triple point at $t=0.2$ with $960 \times 240$ (left) and $1920\times 480$ (right) cells. 30 uniform contours are displayed
  • Figure 5: Laminar boundary layer: the rectangular mesh.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Remark 4.1
  • Remark 4.2