A Compact Third-Order Gas-Kinetic Scheme for Compressible Euler and Navier-Stokes Equations (Communications in Computational Physics, 2020)
In this paper, a compact third-order gas-kinetic scheme is proposed for thecompressible Euler and Navier-Stokes equations. The main reason for the feasibilityto develop such a high-order scheme with compact stencil, which involves onlyneighboring cells, is due to the use of a high-order gas evolution model. Besides theevaluation of the time-dependent flux function across a cell interface, the high-ordergas evolution model also provides an accurate time-dependent solution of the flowvariables at a cell interface. Therefore, the current scheme not only updates the cellaveraged conservative flow variables inside each control volume, but also tracks theflow variables at the cell interface at the next time level. As a result, with both cell averagedand cell interface values, the high-order reconstruction in the current schemecan be done compactly. Different from using a weak formulation for high-order accuracyin the Discontinuous Galerkin method, the current scheme is based on the strongsolution, where the flow evolution starting from a piecewise discontinuous high-orderinitial data is precisely followed. The cell interface time-dependent flow variables canbe used for the initial data reconstruction at the beginning of next time step. Even withcompact stencil, the current scheme has third-order accuracy in the smooth flow regions,and has favorable shock capturing property in the discontinuous regions. It canbe faithfully used from the incompressible limit to the hypersonic flow computations,and many test cases are used to validate the current scheme. In comparison with manyother high-order schemes, the current method avoids the use of Gaussian points forthe flux evaluation along the cell interface and the multi-stage Runge-Kutta time steppingtechnique. Due to its multidimensional property of including both derivatives offlow variables in the normal and tangential directions of a cell interface, the viscousflow solution, especially those with vortex structure, can be accurately captured. Withthe same stencil of a second order scheme, numerical tests demonstrate that the currentscheme is as robust as well-developed second-order shock capturing schemes, butprovides more accurate numerical solutions than the second order counterparts.