Pseudoparticle representation and positivity analysis of explicit and implicit Steger-Warming FVS schemes

Abstract

This paper is about the pseudo-particle representation and the positivity analysis of an explicit and an implicit Steger and Warming's flux vector splitting (FVS) scheme for the compressible Euler equations. The positivity proof is based on the motion of pseudo-particles. For the explicit scheme, it shows that the density and the internal energy could keep non-negative values under the CFL-like condition for the Steger-Warming FVS scheme once the initial gas stays in a physically realizable state. For the implicit method, under a stronger CFL-like condition, the positivity property can also be preserved. Mathematics Subject Classification (2000). 35L65, 65M06, 76M20, 76K05.

A Boltzmann scheme with physically relevant discrete velocities for Euler equations

“…As remarked earlier, we note that the above set of equilibrium distribution functions are similar to the vectors of mass, momentum and energy in the three-particle proposition by Tang and Xu [19] for Steger-Warming flux vectorsplitting scheme [21] for 1-D Euler equations. But it may be noted that the elements corresponding to internal energy are different.…”

“…We thus obtain the discrete velocities which are physically more relevant in line with the wave speeds of the Euler system. The discrete velocities in our formulation are similar to the particle speeds used by Tang and Xu [19] while our derivation and framework are distinctly different.…”

A Gas-Kinetic BGK Scheme for the Navier–Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method

“…An earlier version of the above KFVS scheme is the beam scheme, where instead of using Maxwellians, the equilibrium states g l and g r are replaced by three Delta functions or particles [35]. As analyzed recently [38], the Steger-Warming method can be represented as a "beam scheme" also. However, because of their slight difference in the particle representation, such as the lack of internal energy in the second "particle" in the Steger-Warming method [38], it could be less robust than the beam scheme.…”

“…As analyzed recently [38], the Steger-Warming method can be represented as a "beam scheme" also. However, because of their slight difference in the particle representation, such as the lack of internal energy in the second "particle" in the Steger-Warming method [38], it could be less robust than the beam scheme. The relation between the beam scheme and the Lattice Boltzmann method is analyzed in [50].…”

Steger-Warming flux vector splitting method for special relativistic hydrodynamics

“…This pioneering work has been followed by numerous applications as well as by increasing research efforts to improve further the technique. For example, Tang and Xu in gave a pseudo‐particle representation and the positivity analysis of the Steger–Warming (SW) flux vector splitting (FVS) scheme.…”