POSITIVITY PRESERVING ANALYSIS OF CENTRAL SCHEMES FOR COMPRESSIBLE EULER EQUATIONS

Abstract

Physical quantities like density (ρ), pressure (p), and energy (e) are always non-negative. It is one of the essential qualities a scheme (numerical solver) to maintain the positiveness of these quantities in obtaining the solution of the Euler's equations (or diffrential equations) for inviscid fluid flow. The central solvers add explicitly numerical dissipation to obtain the solution. In this research work, the minimum amount of numerical disspation requirement is analytically obtained for 1D compressible Euler equations by enforcing the positivity criterion to have realistic density, pressure, and internal energy. The positivity analysis of explicit central solver is carried out analytically in 1D for compressible Euler equations to have realistic density, pressure, and energy under the Courant-Friedrichs-Lewy (CFL) condition with Courant number 1. The minimum numerical dissipation criterion obtained in 1D Euler's equations for an explicit solver is a function of the parameters, such as fluid velocity, flow Mach number, and specific heat ratio (γ). The Lax and Friedrichs (L-F) scheme always provides real physical solutions because it has more numerical dissipation than the minimum numerical dissipation required to satisfy positivity. A new positivity-preserving numerical scheme is developed for compressible Euler's equations based on the minimum numerical dissipation in the finite volume framework and tested on the standard test cases in 1D and 2D.