In this presentation, we introduce a flux vector splitting approach to solve the shallow water equations in one and two spatial dimensions. The original ideas stem from the work of Toro and V´azquez on the Euler equations [2]. This approach splits the full system into two processes, referred to as the advection system and the pressure system, and has been implemented to solve the Euler system with a general equation of state[3], the Baer-Nunziato multiphase models, and the shallow water equations [1, 4], among others. After decomposing the conservative flux into advection and pressure components, we construct a Riemann solver for the pressure system based on the system’s eigenstructure. This Riemann solver can be computed either exactly using an inexpensive iterative process or approximately, for example, under a two-rarefaction assumption. Using this information, the advection flux is selected, resulting in a remarkably simple first-order Godunov upwind method. The proposed scheme can serve as a foundation for constructing high-order numerical methods, such as high-order ADER schemes on two-dimensional unstructured meshes. We evaluate the Riemann solver on a suite of test problems with reference solutions, assessing its precision, robustness, and efficiency. Additionally, for one-dimensional problems, we consider classical initial value problems with exact solutions and steady-state solutions for transcritical flow over a bed bump. For two-dimensional problems, we examine convergence problems with artificial source terms, radially symmetric dam-break problems, and tsunami wave propagation in realistic bathymetric scenarios.