Flux-Balanced Patankar-type Schemes for the Compressible Euler Equations

Abstract

Positivity preservation of key physical quantities in the context of fluid flows, such as density and internal energy, is an essential property of a numerical scheme as otherwise the solution lacks physical relevance and has a not well-defined equation of state. One time integration technique that is capable of preserving the positivity of quantities for every time step size is the Patankar-trick and its variants. However, in the context of the Euler equations of gas dynamics, we wonder whether the Patankar-trick should be applied to the density and total energy equations or only to one of them. In this work, we discuss one drawback of the schemes when blindly applied to every positive conserved variable and additionally point out how to overcome the issue by balancing the involved numerical fluxes correctly. To illustrate our findings, we investigate modified Patankar--Runge--Kutta (MPRK) schemes in the context of the compressible Euler equations with and without stiff source terms. We discover that it is beneficial to only apply the Patankar-trick in the density equation and to balance the remaining numerical fluxes consistently rather than applying the trick also to the energy equation. This leads also to the preservation of contact discontinuities. We perform numerical experiments to demonstrate that the accuracy of the methods is maintained while the performance of our approach is superior to the traditional application of MPRK schemes.

Figures (3)

• Figure 1: Plots of the numerical solution of multi-species Euler equations \ref{['eq:EulerEquations']}-\ref{['eq:multiEuler']} with source \ref{['eq:source']} using $\delta=10^4$, $N=4000$ cells as well as a CFL and safety factor as given in Table \ref{['tab:react_euler']}. First-order methods are at the left, second-order methods are at the right.
• Figure 2: Plots of the numerical solution of Euler equations \ref{['eq:EulerEquations']}-\ref{['eq:monoEuler']} with contact discontinuity IC \ref{['eq:cont_disc']} with $N=1000$ using a CFL and safety factor given in Table \ref{['tab:cont_disc']}. First-order methods are left, second-order methods are on the right.
• Figure 3: Plots of the numerical solution of Euler equations \ref{['eq:EulerEquations']}-\ref{['eq:monoEuler']} with vacuum IC \ref{['eq:vacuum']} with $N=1000$, and with CFL and safety factor given in Table \ref{['tab:vacuum']}. First-order methods are left, second-order methods are on the right.

Theorems & Definitions (12)

• Definition 1
• Remark 1
• Lemma 1
• proof
• Proposition 2: Contact discontinuity preservation of explicit Euler LLF
• proof
• Theorem 3
• proof
• Theorem 4
• proof