High-order well-balanced central WENO scheme for pre-balanced shallow water equations

In this paper, we are concerned with numerically solving shallow water equations with a source term. The equations admit steady state solutions in which the non-zero flux gradient is exactly balanced by the source term. It is difficult for numerical schemes to preserve the delicate balance between the flux gradient and the source term. By virtue of a modified governing equations as well as an original reconstruction procedure for the Runge–Kutta flux, we present a central weighted essentially non-oscillatory scheme. Extensive test cases are carried out to demonstrate the performance of the resulting scheme: exact conservation property, high-order accuracy and good resolution near discontinuities.