Seventh order Hermite WENO scheme for hyperbolic conservation laws

In this paper, we construct a new seventh-order Hermite weighted essentially non-oscillatory (HWENO7) scheme, for solving one and two dimensional hyperbolic conservation laws, by extending the fifth order HWENO introduced in Qiu J, Shu C-W. J Comput Phys 2003;193:115-135. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both function and its 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. The major advantage of HWENO schemes is its compactness in the reconstruction. For example, seven points are needed in the stencil for a seventh order WENO reconstruction, while only five points are needed for HWENO7 reconstruction. We use the central-upwind flux [Kurganov A, Noelle S, Petrova G, SIAM J Sci Comp 2001;23:707-740] which is simple, universal and efficient. The numerical solution is advanced in time by the seventh order linear strong-stability-preserving Runge-Kutta (ℓSSPRK) scheme for linear problems and the fourth order SSPRK(5,4) for nonlinear problems. The resulting scheme improves the convergence and accuracy of smooth parts of solution as well as decrease the dissipation near discontinuities. This is especially for long time evolution problems. Numerical experiments of the new scheme for one and two dimensional problems are reported. The results demonstrate that the proposed scheme is superior to the original HWENO and classical WENO schemes.