High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations (Communications in Computational Physics, 2017)
In this study, a numerical framework of the high order well-balancedweighted compact nonlinear (WCN) schemes is proposed for the shallow water equationsbased on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008)7294-7321]. We employ a special splitting technique for the source term proposed in[Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property,which can be proved theoretically. In the meantime, the genuine high orderaccuracy of the numerical scheme can be observed successfully, and small perturbationof the stationary state can be resolved and evolved well. In order to capture thestrong discontinuities and large gradients, the fifth-order upwind weighted nonlinearinterpolations together with the fourth/sixth order cell-centered compact scheme areused to construct different WCN schemes. In addition, the local characteristic projectionsare considered to further restrain the potential numerical oscillations. A varietyof representative one- and two-dimensional examples are tested to demonstrate thegood performance of the proposed schemes.