On conditions for weak conservativeness of regularized explicit finite-difference schemes for 1D barotropic gas dynamics equations

TL;DR

This paper derives necessary and sufficient conditions for the weak conservativeness of explicit finite-difference schemes applied to 1D barotropic gas dynamics, ensuring stability and accuracy in numerical simulations.

Contribution

It introduces a precise von Neumann type criterion for weak conservativeness of regularized schemes, improving upon previous conditions and aligning with numerical results.

Findings

Derived a necessary and sufficient CFL-type criterion for weak conservativeness.

The criterion is narrower than the necessary condition and broader than the previous sufficient condition.

Numerical results confirm the criterion's accuracy for the original gas dynamics system.

Abstract

We consider explicit two-level three-point in space finite-difference schemes for solving 1D barotropic gas dynamics equations. The schemes are based on special quasi-gasdynamic and quasi-hydrodynamic regularizations of the system. We linearize the schemes on a constant solution and derive the von Neumann type necessary condition and a CFL type criterion (necessary and sufficient condition) for weak conservativeness in $L^2$ for the corresponding initial-value problem on the whole line. The criterion is essentially narrower than the necessary condition and wider than a sufficient one obtained recently in a particular case; moreover, it corresponds most well to numerical results for the original gas dynamics system.