A kinetic flux-vector splitting method for single-phase and two-phase shallow flows

A high order kinetic flux-vector splitting method (KFVS) is applied to solve single-phase and two-phase shallow flow equations. The single-phase shallow water equations contain the flow height and momentum. On the other hand, the two-phase flow is considered as a shallow layer of solid granular material and fluid over a horizontal surface. The flow components are assumed to be incompressible and the flow height, solid volume fraction and phase momenta are considered. Our interest lies in the numerical approximation of the above mentioned models, whose complexities pose numerical difficulties. The proposed numerical method is based on the direct splitting of macroscopic flux functions of the system of equations. The two-phase shallow flow model governs a non-homogeneous conservation law and, thus, the scheme is extended to account for the non homogeneous cases. The higher order accuracy of the scheme is achieved by using a MUSCL-type initial reconstruction and the Runge-Kutta time stepping method. A number of numerical test problems are considered. For validation, the results of the proposed method are compared with those obtained from the staggered central scheme. The numerical results show the accuracy and robustness of the suggested solver.