# Preprint 2010-019

# Comparison of the generalized Riemann solver and the gas-kinetic scheme for inviscid compressible flow simulations

## Jiequan Li, Qibing Li and Kun Xu

**Abstract:**
The generalized Riemann problem (GRP) scheme
for the Euler equations
and gas-kinetic scheme (GKS) for the Boltzmann equation
are two high resolution shock capturing schemes
for fluid simulations.
The difference is that one is based on the characteristics
of the inviscid Euler equations and their wave interactions,
and the other is based on the particle transport and collisions.
The similarity between them is that both methods
can use identical MUSCL-type initial reconstructions
around a cell interface,
and the spatial slopes on both sides of a cell interface
involve in the gas evolution process and
the construction of a time-dependent flux function.
Although both methods have been applied successfully
to the inviscid compressible flow computations,
their performances have never been compared.
Since both methods use the same initial reconstruction,
any difference is solely coming from
different underlying mechanism in their flux evaluation.
Therefore, such a comparison is important to help us
to understand the correspondence
between physical modeling and numerical performances.
Since GRP is so faithfully solving the inviscid Euler equations,
the comparison can be also used to show the
validity of solving the Euler equations itself.
The numerical comparison shows that the GRP exhibits
a better computational efficiency,
and has comparable accuracy with GKS for the Euler solutions
in 1D case, but the GKS is more robust than GRP.
For the 2D high Mach number flow simulations,
the GKS is absent from the shock instability
and converges to the steady state solutions faster than the GRP.
The GRP has carbuncle phenomena, likes a cloud
hanging over exact Riemann solver.
The GRP and GKS use different physical processes
to describe the flow motion starting from a discontinuity.
One is based on the assumption of equilibrium state
with infinite number of particle collisions,
and the other starts from
the non-equilibrium free transport process
to evolve into an equilibrium one through particle collisions.
The different mechanism in the flux evaluation
deviates their performance.
Through this study, we may conclude scientifically
that it is NOT valid to use the Euler equations
to construct numerical fluxes in a discretized space.
To adapt Navierâ€“Stokes (NS) equations is NOT valid either because
the NS has no any account on the strength of the discontinuity.
A direct modeling of the physical process in the discretized
space is necessary in the construction of numerical scheme.
This process is similar to the modeling
in deriving the governing equations,
but the control volume here cannot be shrunk to zero.