Hybrid finite difference WENO schemes for the ten-moment Gaussian closure equations with source term

A hybrid weighted essentially non-oscillatory (WENO) finite difference scheme is proposed for computing discontinuous solutions of the ten-moment Gaussian closure equations. A salient feature of the proposed scheme is the use of low-cost component-wise reconstruction of the numerical fluxes in smooth regions and non-oscillatory characteristic-wise reconstruction in the vicinity of discontinuities. A troubled-cell indicator which measures the smoothness of the solution, and built on utilising the smoothness indicators of the underlying WENO scheme, is employed to effectively switch between the two reconstructions. The resulting hybrid WENO scheme is simple and efficient, is independent of the order and type of the WENO reconstruction, and it can be used as an effective platform to construct finite difference schemes of any arbitrary high-order accuracy. For demonstration, we have considered the fifth order WENO-Z reconstruction. We have performed several 1D and 2D numerical experiments to illustrate the efficiency of the proposed hybrid algorithm and its performance compared to the standard WENO-Z scheme. Numerical case studies shows that the present algorithm achieves fifth order accuracy for smooth problems, resolves discontinuities in a non-oscillatory manner and takes 25%–50% less computational time than the WENO-Z scheme while retaining many of its advantages.