On the Construction, Comparison, and Local Characteristic Decomposition for High-Order Central WENO Schemes

Abstract

In this paper, we review and construct fifth- and ninth-order central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear weights appear in such a formulation and they are treated using the technique recently introduced by Shi et al. ( J. Comput. Phys. 175, 108 (2002)). We then perform numerical computations and comparisons with the finite difference WENO schemes of Jiang and Shu ( J. Comput. Phys. 150, 97 (1999)) and Balsara and Shu ( J. Comput. Phys. 160, 405 (2000)). 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 control spurious oscillations when the order of accuracy is high, both for the central staggered grid and for the upwind nonstaggered grid WENO schemes. We use the shock entropy wave interaction problem to demonstrate the advantage of using higher order WENO schemes when both shocks and complex solution features coexist.