Finite Difference Hermite WENO Schemes for Hyperbolic Conservation Laws

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one and two dimensional nonlinear hyperbolic conservation law systems is presented. The construction of HWENO schemes is based on a finite difference formulation, Hermite interpolation, and nonlinearly stable Runge–Kutta methods. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. Comparing with the original finite difference WENO schemes of Jiang and Shu (J Comput Phys 126:202–228, 1996), one major advantage of HWENO schemes is its compactness in the reconstruction. For example, five points are needed in the stencil for a fifth order WENO (WENO5) reconstruction, while only three points are needed for a fifth order HWENO (HWENO5) reconstruction. Some benchmark numerical experiments are presented to illustrate efficiency of HWENO schemes.