Flux-vector splitting is a method used in numerical approximations of hyperbolic systems of conservation laws, such as the Euler equations. The goal is to split the flux $ f(w) $ into a forward flux $ f^{+}(w) $ and a backward flux $ f^{-}(w) $, ensuring that the derivatives of these fluxes have eigenvalues with specific signs. This splitting is crucial for upwind differencing, which requires knowing the direction of wave propagation in the computational grid. Two physical models are used for upwind differencing: the Riemann approach and the Boltzmann approach. The Riemann approach involves solving Riemann's initial-value problem to determine wave characteristics, while the Boltzmann approach uses a particle-based model. Flux-difference splitting and flux-vector splitting are numerical techniques derived from these models. Flux-vector splitting ensures that the forward flux has non-negative eigenvalues and the backward flux has non-positive eigenvalues. The splitting must satisfy several conditions, including continuity of the fluxes, symmetry with respect to the Mach number, and continuity of their derivatives. Additionally, the splitting must result in a polynomial in the Mach number of the lowest possible degree. The paper focuses on flux-vector splitting for the Euler equations with the ideal-gas law as the equation of state. The method ensures that in supersonic regions, flux-vector splitting leads to standard upwind differencing. The symmetry and continuity conditions ensure that the numerical results are smooth and accurate. The requirement that the derivatives of the fluxes have one eigenvalue vanish for subsonic regions allows for the construction of stationary shock structures with minimal interior zones. Finally, the polynomial nature of the fluxes ensures the uniqueness of the splitting.Flux-vector splitting is a method used in numerical approximations of hyperbolic systems of conservation laws, such as the Euler equations. The goal is to split the flux $ f(w) $ into a forward flux $ f^{+}(w) $ and a backward flux $ f^{-}(w) $, ensuring that the derivatives of these fluxes have eigenvalues with specific signs. This splitting is crucial for upwind differencing, which requires knowing the direction of wave propagation in the computational grid. Two physical models are used for upwind differencing: the Riemann approach and the Boltzmann approach. The Riemann approach involves solving Riemann's initial-value problem to determine wave characteristics, while the Boltzmann approach uses a particle-based model. Flux-difference splitting and flux-vector splitting are numerical techniques derived from these models. Flux-vector splitting ensures that the forward flux has non-negative eigenvalues and the backward flux has non-positive eigenvalues. The splitting must satisfy several conditions, including continuity of the fluxes, symmetry with respect to the Mach number, and continuity of their derivatives. Additionally, the splitting must result in a polynomial in the Mach number of the lowest possible degree. The paper focuses on flux-vector splitting for the Euler equations with the ideal-gas law as the equation of state. The method ensures that in supersonic regions, flux-vector splitting leads to standard upwind differencing. The symmetry and continuity conditions ensure that the numerical results are smooth and accurate. The requirement that the derivatives of the fluxes have one eigenvalue vanish for subsonic regions allows for the construction of stationary shock structures with minimal interior zones. Finally, the polynomial nature of the fluxes ensures the uniqueness of the splitting.