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Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations

Saurav Samantaray

Abstract

We consider the compressible Euler equations of gas dynamics with isentropic equation of state. Standard numerical schemes for the Euler equations suffer from stability and accuracy issues in the low Mach regime. These failures are attributed to the transitional behaviour of the governing equations from compressible to incompressible solution in the limit of vanishing Mach number. In this paper we introduce an extra flux term to the momentum flux. This extra term is recognised by looking at the constraints of the incompressible limit system. As a consequence the flux terms enable us to get a suitable splitting, so that an additive IMEX-RK scheme could be applied. Using an elliptic reformulation the scheme boils down to just solving a linear elliptic problem for the density and then explicit updates for the momentum. The IMEX schemes developed are shown to be formally asymptotically consistent with the low Mach number limit of the Euler equations and are shown to be linearly $L^2$ stable. A second order space time fully discrete scheme is obtained in the finite volume framework using a combination of Rusanov flux for the explicit part and simple central differences for the implicit part. Results of numerical case studies are reported which elucidate the theoretical assertions regarding the scheme and its robustness.

Asymptotic Preserving Linearly Implicit Additive IMEX-RK Finite Volume Schemes for Low Mach Number Isentropic Euler Equations

Abstract

We consider the compressible Euler equations of gas dynamics with isentropic equation of state. Standard numerical schemes for the Euler equations suffer from stability and accuracy issues in the low Mach regime. These failures are attributed to the transitional behaviour of the governing equations from compressible to incompressible solution in the limit of vanishing Mach number. In this paper we introduce an extra flux term to the momentum flux. This extra term is recognised by looking at the constraints of the incompressible limit system. As a consequence the flux terms enable us to get a suitable splitting, so that an additive IMEX-RK scheme could be applied. Using an elliptic reformulation the scheme boils down to just solving a linear elliptic problem for the density and then explicit updates for the momentum. The IMEX schemes developed are shown to be formally asymptotically consistent with the low Mach number limit of the Euler equations and are shown to be linearly stable. A second order space time fully discrete scheme is obtained in the finite volume framework using a combination of Rusanov flux for the explicit part and simple central differences for the implicit part. Results of numerical case studies are reported which elucidate the theoretical assertions regarding the scheme and its robustness.
Paper Structure (21 sections, 4 theorems, 68 equations, 13 figures, 9 tables)

This paper contains 21 sections, 4 theorems, 68 equations, 13 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Euler Equations Scaled With Low Mach Regime References
  3. Zero Mach Number Limit $\varepsilon \to 0$
  4. Time Discretisation
  5. Equilibrium Reformulation
  6. First Order Time Semi-discrete Scheme
  7. Asymptotic Consistency
  8. Asymptotic Stability
  9. Higher Order Extension of Time Discretisation
  10. IMEX-RK Time Discretisation
  11. Higher Order Time Discretisation for the Euler Equations
  12. Asymptotic Consistency
  13. Asymptotic Stability
  14. Space-Time Fully Discrete Finite Volume IMEX-RK Schemes
  15. Numerical Results
  16. ...and 6 more sections

Key Result

Proposition 3.5

The first-order time semi-discrete scheme in Definition def:1st_TSD and the reformulated first-order time semi-discrete scheme in Definition def:1st_RTSD, are equivalent.

Figures (13)

  • Figure 1: Double Butcher tableau of an IMEX-RK scheme.
  • Figure 1: Double Butcher tableaux of type-A Additive IMEX schemes. DP2-A(2, 4, 2).
  • Figure 2: 1D Riemann Problem: Density and momentum profiles at time T = 0.05, in the compressible regime i.e. $\varepsilon = 0.8 \hbox{and} 0.3$ for the AP and CL scheme for N = 200
  • Figure 3: 1D Riemann Problem: Density and momentum profiles at time T = 0.05, in the weakly compressible and incompressible regime i.e. $\varepsilon = 0.05 \hbox{and} 0.005$, respectively, for the AP and CL scheme for N = 200
  • Figure 4: Colliding Pulses: Plots of density profiles for the AP scheme at time various times for $\varepsilon = 0.1$, N = 200
  • ...and 8 more figures

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proof 1
  • Remark 3.7
  • ...and 14 more