Improved ninth order WENO scheme for hyperbolic conservation laws

In this paper, we construct a new ninth order WENO scheme by extending the improved fifth order WENO introduced in [R. Borges, M. Carmona, B. Costa, W. Sun Don, J. Comput. Phys. 227 (2008) 3191–3211] (IWENO). We use the central-upwind flux [A. Kurganov, S. Noelle, G. Petrova, SIAM J. Sci. Comp. 23 (2001) 707–740] which is simple, universal and efficient. The numerical solution is advanced in time by the ninth order linear strong-stability-preserving Runge–Kutta (ℓSSPRK) scheme. The resulting scheme improves the convergence and accuracy at critical points 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 IWENO and classical WENO schemes.