Comparison of Fifth-Order WENO Scheme and Finite Volume WENO-Gas-Kinetic Scheme for Inviscid and Viscous Flow Simulation (Communications in Computational Physics, 2013)
The development of high-order schemes has been mostly concentrated onthe limiters and high-order reconstruction techniques. In this paper, the effect of theflux functions on the performance of high-order schemes will be studied. Based on thesame WENO reconstruction, two schemes with different flux functions, i.e., the fifth-order WENO method and the WENO-Gas-Kinetic scheme (WENO-GKS), will be compared. The fifth-order finite difference WENO-SW scheme is a characteristic variablereconstruction based method which uses the Steger-Warming flux splitting for inviscid terms, the sixth-order central difference for viscous terms, and three stages Runge-Kutta time stepping for the time integration. On the other hand, the finite volumeWENO-GKS is a conservative variable reconstruction based method with the sameWENO reconstruction. But it evaluates a time dependent gas distribution functionalong a cell interface, and updates the flow variables inside each control volume byintegrating the flux function along the boundary of the control volume in both spaceand time. In order to validate the robustness and accuracy of the schemes, both methods are tested under a wide range of flow conditions: vortex propagation, Mach 3step problem, and the cavity flow at Reynolds number 3200. Our study shows thatboth WENO-SW and WENO-GKS yield quantitatively similar results and agree witheach other very well provided a sufficient grid resolution is used. With the reduction of mesh points, the WENO-GKS behaves to have less numerical dissipation andpresent more accurate solutions than those from the WENO-SW in all test cases. Forthe Navier-Stokes equations, since the WENO-GKS couples inviscid and viscous termsin a single flux evaluation, and the WENO-SW uses an operator splitting technique, itappears that the WENO-SW is more sensitive to the WENO reconstruction and boundary treatment. In terms of efficiency, the finite volume WENO-GKS is about 4 timesslower than the finite difference WENO-SW in two dimensional simulations. The current study clearly shows that besides high-order reconstruction, an accurate gas evolution model or flux function in a high-order scheme is also important in the capturing of physical solutions. In a physical flow, the transport, stress deformation, heat conduction, and viscous heating are all coupled in a single gas evolution process. Therefore,it is preferred to develop such a scheme with multi-dimensionality, and unified treatment of inviscid and dissipative terms. A high-order scheme does prefer a high-ordergas evolution model. Even with the rapid advances of high-order reconstruction techniques, the first-order dynamics of the Riemann solution becomes the bottleneck forthe further development of high-order schemes. In order to avoid the weakness of thelow order flux function, the development of high-order schemes relies heavily on theweak solution of the original governing equations for the update of additional degreeof freedom, such as the non-conservative gradients of flow variables, which cannot bephysically valid in discontinuous regions.