An Improved Seventh Order Hermite WENO Scheme for Hyperbolic Conservation Laws

Abstract

A seventh order Hermite weighted essentially non-oscillatory (HWENO) scheme [¹] has been used successfully to solve hyperbolic conservation laws. However, the scheme could not achieve the optimal order of accuracy at critical points. In this paper, we propose a new central seventh order HWENO scheme. It involves seventh order HWENO reconstruction [¹] and the central-upwind flux [^{2, 3}]. For the ideal weights we used the central procedure presented in [⁴] and for nonlinear weights we extend the strategy proposed in [^{5, 6}]. For time integration, the seventh order linear strong-stability-preserving Runge–Kutta (ℓSSPRK) scheme is used. The resulting scheme combines the advantages of both the improved and Hermite central schemes, e.g., compactness, improving the convergence and accuracy at critical points, decreasing the dissipation near discontinuities especially for long time evolution problems; it is simple to implement and attains seventh order accuracy. Many numerical tests are presented to validate the performance of the proposed scheme.