An upwind second-order scheme for compressible duct flows
SIAM Journal on Scientific and Statistical Computing
The generalized Riemann problem for reactive flows
Journal of Computational Physics
SIAM Journal on Mathematical Analysis
On Godunov-type methods near low densities
Journal of Computational Physics
Numerical solution of the Riemann problem for two-dimensional gas dynamics
SIAM Journal on Scientific Computing
A singularities tracking conservation laws scheme for compressible duct flows
Journal of Computational Physics
A two-dimensional conservation laws scheme for compressible flows with moving boundaries
Journal of Computational Physics
Numerical Instablilities in Upwind Methods: Analysis and Cures for the “Carbuncle” Phenomenon
Journal of Computational Physics
Journal of Computational Physics
On the construction of kinetic schemes
Journal of Computational Physics
A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow
Journal of Computational Physics
A compressible Navier-Stokes flow solver with scalar transport
Journal of Computational Physics
A note on the conservative schemes for the Euler equations
Journal of Computational Physics
Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem
Numerische Mathematik
Journal of Computational Physics
New lax-friedrichs scheme for convective-diffusion equation
ICICA'12 Proceedings of the Third international conference on Information Computing and Applications
A new gas-kinetic scheme based on analytical solutions of the BGK equation
Journal of Computational Physics
The adaptive GRP scheme for compressible fluid flows over unstructured meshes
Journal of Computational Physics
Hi-index | 31.46 |
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 slightly 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 solvers. 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 numerical performance. Through this study, we may conclude scientifically that it may NOT be valid to use the Euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution. To adapt the Navier-Stokes (NS) equations is NOT valid either because the NS equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity. This fact alludes to the consistency of the Euler and Navier-Stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high Mach number flows. The development of numerical algorithm is similar to the modeling process in deriving the governing equations, but the control volume here cannot be shrunk to zero.