A Compact Third-order Gas-kinetic Scheme for Compressible Euler and Navier-Stokes Equations

摘要

In this paper, a compact third-order gas-kinetic scheme is proposed for thecompressible Euler and Navier-Stokes equations. The main reason for thefeasibility to develop such a high-order scheme with compact stencil, whichinvolves only neighboring cells, is due to the use of a high-order gasevolution model. Besides the evaluation of the time-dependent flux functionacross a cell interface, the high-order gas evolution model also provides anaccurate time-dependent solution of the flow variables at a cell interface.Therefore, the current scheme not only updates the cell averaged conservativeflow variables inside each control volume, but also tracks the flow variablesat the cell interface at the next time level. As a result, with both cellaveraged and cell interface values the high-order reconstruction in the currentscheme can be done compactly. Different from using a weak formulation forhigh-order accuracy in the Discontinuous Galerkin (DG) method, the currentscheme is based on the strong solution, where the flow evolution starting froma piecewise discontinuous high-order initial data is precisely followed. Thecell interface time-dependent flow variables can be used for the initial datareconstruction at the beginning of next time step. Even with compact stencil,the current scheme has third-order accuracy in the smooth flow regions, and hasfavorable shock capturing property in the discontinuous regions. Many testcases are used to validate the current scheme. In comparison with many otherhigh-order schemes, the current method avoids the use of Gaussian points forthe flux evaluation along the cell interface and the multi-stage Runge-Kuttatime stepping technique.