On the construction comparison and local characteristic decomposition for higher order central WENO schemes

In this paper, we first construct fourth and eighth order central WENO (weighted essen-tially non-oscillatory) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge-Kutta methods, for solving nonlinear hyperbolic conserva-tion law systems. Negative linear weights appear in such a formulation and they are treated using the technique recently introduced by Shi, Hu and Shu [19]. We then perform numer-ical computations and comparisons with finite difference WENO schemes of Jiang and Shu [5] and Balsara and Shu [1]. The emphasis is on the performance with or without a local characteristic decomposition. While this decomposition increases the computational cost, we demonstrate by our numerical experiments that it is still necessary to use it to remove spurious oscillations when the order of accuracy is high, both for the central staggered grid and for the upwind non-staggered grid WENO schemes. We use the shock entropy wave interaction problem to demonstrate the advantage of using a higher order WENO schemes when both shocks and complex solution features co-exist. Key Words: WENO scheme, central scheme, high order accuracy, local characteristic decomposition