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Stability evaluation of approximate Riemann solvers using the direct Lyapunov method

Aishwarjya Gogoi, Jadav Chandra Mandal, Amitabh Saraf

Abstract

The paper presents a new approach of stability evaluation of the approximate Riemann solvers based on the direct Lyapunov method. The present methodology offers a detailed understanding of the origins of numerical shock instability in the approximate Riemann solvers. The pressure perturbation feeding the density and transverse momentum perturbations is identified as the cause of the numerical shock instabilities in the complete approximate Riemann solvers, while the magnitude of the numerical shock instabilities are found to be proportional to the magnitude of the pressure perturbations. A shock-stable HLLEM scheme is proposed based on the insights obtained from this analysis about the origins of numerical shock instability in the approximate Riemann solvers. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems of the original HLLEM scheme at high Mach numbers.

Stability evaluation of approximate Riemann solvers using the direct Lyapunov method

Abstract

The paper presents a new approach of stability evaluation of the approximate Riemann solvers based on the direct Lyapunov method. The present methodology offers a detailed understanding of the origins of numerical shock instability in the approximate Riemann solvers. The pressure perturbation feeding the density and transverse momentum perturbations is identified as the cause of the numerical shock instabilities in the complete approximate Riemann solvers, while the magnitude of the numerical shock instabilities are found to be proportional to the magnitude of the pressure perturbations. A shock-stable HLLEM scheme is proposed based on the insights obtained from this analysis about the origins of numerical shock instability in the approximate Riemann solvers. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems of the original HLLEM scheme at high Mach numbers.
Paper Structure (23 sections, 50 equations, 10 figures, 1 table)

This paper contains 23 sections, 50 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Finite volume discretization
  4. HLL family flux function
  5. The HLL approximate Riemann solver
  6. The HLLEM approximate Riemann solver
  7. Linear perturbation analysis
  8. Refined analysis of HLL-type schemes
  9. Reduced or first Lyapunov method for discrete Euler equations
  10. Direct Lyapunov method
  11. Global asymptotic stability criterion
  12. Analysis of the HLLE scheme
  13. Analysis of the Roe, HLLEM and HLLC schemes
  14. Analysis of the HLLCM and HLLEC schemes
  15. Analysis of the HLLS and HLLES schemes
  16. ...and 8 more sections

Figures (10)

  • Figure 1: Density contours for $M_{\infty}=20$ flow over a blunt body computed by the HLLEM and HLLEM-LM schemes
  • Figure 2: Phase portrait of the HLLEM and proposed HLLEM-FP1D schemes for positive pressure and negative density perturbations
  • Figure 3: Density contours for the $M_{\infty}=6$ planar shock problem computed by the HLLE, HLLEM and HLLEM-FP1D schemes. The results are shown at time $t=55$ units.
  • Figure 4: Density contours for the $M_{\infty}=10$ double Mach reflection problem computed by the first-order HLLE, HLLEM and HLLEM-FP1D schemes. The results are shown at time $t=0.020026$ units.
  • Figure 5: Density contours for the $M_{\infty}=10$ double Mach reflection problem computed by the second-order HLLE and HLLEM-FP1D schemes. The results are shown at time $t=0.020026$ units.
  • ...and 5 more figures