2 research outputs found
A HWENO Reconstruction Based High-order Compact Gas-kinetic Scheme on Unstructured Meshes
As an extension of previous fourth-order compact gas kinetic scheme (GKS) on
structured meshes (Ji et al. 2018), this work is about the development of a
third-order compact GKS on unstructured meshes for the compressible Euler and
Navier-Stokes solutions. Based on the time accurate high-order gas-kinetic
evolution solution at a cell interface, the time dependent gas distribution
function in GKS provides not only the flux function and its time derivative at
a cell interface, but also the time accurate flow variables there at next time
level. As a result, besides updating the conservative flow variables inside
each control volume through the interface fluxes, the cell averaged first-order
spatial derivatives of flow variables in the cell can be also obtained using
the updated flow variables at the cell interfaces around that cell through the
divergence theorem.
Therefore, with the flow variables and their first-order spatial derivatives
inside each cell, the Hermite WENO (HWENO) techniques can be naturally
implemented for the compact high-order reconstruction at the beginning of a new
time step.
Following the reconstruction method in (Zhu et al. 2018), a new HWENO
reconstruction on triangular meshes is designed in the current scheme. Combined
with a two-stage temporal discretization and second-order gas-kinetic flux
function, a third-order spatial and temporal accuracy in the current compact
scheme can be achieved. Accurate solutions can be obtained for both inviscid
and viscous flows without sensitive dependence on the quality of triangular
meshes. The robustness of the scheme has been validated as well through the
cases with strong shocks in the hypersonic viscous flow simulations
Performance Enhancement for High-order Gas-kinetic Scheme Based on WENO-adaptive-order Reconstruction
High-order gas-kinetic scheme (HGKS) has been well-developed in the past
years. Abundant numerical tests including hypersonic flow, turbulence, and
aeroacoustic problems, have been used to validate its accuracy, efficiency, and
robustness. However, there are still rooms for its further improvement.
Firstly, the reconstruction in the previous scheme mainly achieves a
third-order accuracy for the initial non-equilibrium states. At the same time,
the equilibrium state in space and time in HGKS has to be reconstructed
separately. Secondly, it is complicated to get reconstructed data at Gaussian
points from the WENO-type method in high dimensions. For HGKS, besides the
point-wise values at the Gaussian points it also requires the slopes in both
normal and tangential directions of a cell interface. Thirdly, there exists
visible spurious overshoot/undershoot at weak discontinuities from the previous
HGKS with the standard WENO reconstruction. In order to overcome these
difficulties, in this paper we use an improved reconstruction for HGKS. The
WENO with adaptive order (WENO-AO) method is implemented for reconstruction.A
whole polynomial inside each cell is provided in WENO-AO reconstruction. The
HGKS becomes simpler than the previous one with the direct implementation of
cell interface values and their slopes from WENO-AO. The additional
reconstruction of equilibrium state at the beginning of each time step can be
avoided as well by dynamically merging the reconstructed non-equilibrium
slopes. The new HGKS essentially releases or totally removes the above existing
problems in previous HGKS. The accuracy of the scheme from 1D to 3D from the
new HGKS can recover the theoretical order of accuracy of the WENO
reconstruction.In the two- and three-dimensional simulations, the new HGKS
shows better robustness and efficiency than the previous scheme in all test
cases