A new numerical solver is developed for a shallow-water flow model to simulate meandering river flows. This solver can investigate the flows in the curved river channel, by means of the adaptive CIP-Soroban (CIP-S) scheme in a cylindrical coordinate system. Time development of water velocity and free-surface level is computed by the orthogonal curvilinear coordinate system without any transformation of the governing equations, and the advection term is calculated by the high-accuracy CIP scheme. The numerical grid points of the waterfront line at the river bank are moved and tracked by the kinematic condition so that the grid points are always located on the line, and the rearrangement of the positions of the girds is easily conducted by the original CIP-S method. From the verification of this solver by the pure advection problem and by the numerical solution of the meandering river flows, in comparison with the CIP-S method in a Cartesian coordinate system and with the boundary fitted coordinate (BFC) method, it is shown that the proposed numerical solver reasonably predicts the main flow profile and water surface elevation and will be a promising numerical method as one of the practical solutions.