Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Developing high-order weighted compact nonlinear schemes
Journal of Computational Physics
Journal of Computational Physics
Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation
Journal of Computational Physics
Journal of Computational Physics
The kinetic scheme for the full-Burnett equations
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
Spectral difference method for unstructured grids I: basic formulation
Journal of Computational Physics
On the multidimensional gas-kinetic BGK scheme
Journal of Computational Physics
A Runge-Kutta discontinuous Galerkin method for viscous flow equations
Journal of Computational Physics
High order multi-moment constrained finite volume method. Part I: Basic formulation
Journal of Computational Physics
A unified gas-kinetic scheme for continuum and rarefied flows
Journal of Computational Physics
A gas kinetic scheme for the Baer-Nunziato two-phase flow model
Journal of Computational Physics
WENO-enhanced gas-kinetic scheme for direct simulations of compressible transition and turbulence
Journal of Computational Physics
A new gas-kinetic scheme based on analytical solutions of the BGK equation
Journal of Computational Physics
The generalized Riemann problems for compressible fluid flows: Towards high order
Journal of Computational Physics
Hi-index | 31.47 |
The foundation for the development of modern compressible flow solver is based on the Riemann solution of the inviscid Euler equations. The high-order schemes are basically related to high-order spatial interpolation or reconstruction. In order to overcome the low-order wave interaction mechanism due to the Riemann solution, the temporal accuracy of the scheme can be improved through the Runge-Kutta method, where the dynamic deficiencies in the first-order Riemann solution is alleviated through the sub-step spatial reconstruction in the Runge-Kutta process. The close coupling between the spatial and temporal evolution in the original nonlinear governing equations seems weakened due to its spatial and temporal decoupling. Many recently developed high-order methods require a Navier-Stokes flux function under piece-wise discontinuous high-order initial reconstruction. However, the piece-wise discontinuous initial data and the hyperbolic-parabolic nature of the Navier-Stokes equations seem inconsistent mathematically, such as the divergence of the viscous and heat conducting terms due to initial discontinuity. In this paper, based on the Boltzmann equation, we are going to present a time-dependent flux function from a high-order discontinuous reconstruction. The theoretical basis for such an approach is due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the kinetic equation has the mechanism to construct a dissipative wave structure starting from an initially discontinuous flow condition on a time scale being larger than the particle collision time. The current high-order flux evaluation method is an extension of the second-order gas-kinetic BGK scheme for the Navier-Stokes equations (BGK-NS). The novelty for the easy extension from a second-order to a higher order is due to the simple particle transport and collision mechanism on the microscopic level. This paper will present a hierarchy to construct such a high-order method. The necessity to couple spatial and temporal evolution nonlinearly in the flux evaluation can be clearly observed through the numerical performance of the scheme for the viscous flow computations.