HWENO Schemes Based on Compact Differencefor Hyperbolic Conservation Laws

In this paper, in order to simplify the high order Hermite weighted essentially non-oscillatory (HWENO) finite difference schemes of Liu and Qiu (J Sci Comput 63:548–572, 2015), a new type of HWENO schemes based on compact difference schemes, termed CHWENO (compact HWENO) schemes, is proposed for solving both one and two dimensional hyperbolic conservation laws. The idea of reconstruction in CHWENO schemes is similar to HWENO schemes, however the first derivative values of solution are solved by the compact difference method and only one numerical flux is used in CHWENO schemes, while the derivative equation is needed to be solved and two numerical fluxes are used in HWENO schemes. Compared with the original finite difference weighted essentially non-oscillatory schemes, CHWENO schemes maintain the compactness of HWENO schemes, which means that only three points are needed for a fifth order CHWENO schemes. Compared with the HWENO schemes, CHWENO schemes avoid solving the complex derivative equations, which can considerably expedite calculating speed. Several numerical tests, including the 1D shock density wave interaction problem and 2D Riemann problem, are presented to demonstrate the efficiency of CHWENO schemes.