Fourth-order balanced source term treatment in central WENO schemes for shallow water equations

The aim of this work is to develop a well-balanced Central Weighted Essentially Non Oscillatory (CWENO) method, fourth-order accurate in space and time, for shallow water system of balance laws with bed slope source term. Time accuracy is obtained applying a Runge-Kutta scheme (RK), coupled with the Natural Continuous Extension (NCE) approach. Space accuracy is obtained using WENO reconstructions of the conservative variables and of the water-surface elevation. Extension of the applicability of the standard CWENO scheme to very irregular bottoms, preserving high order accuracy, is obtained introducing two original procedures. The former involves the evaluation of the point-values of the flux derivative, coupled with the bed source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the regularity of the free surface elevation, usually smoother than the bottom elevation. Both these procedures satisfy the C-property, the property of exactly preserving the quiescent flow. Several standard one-dimensional test cases are used to verify high-order accuracy, exact C-property, and good resolution properties for smooth and discontinuous solutions.