The paper by Bram van Leer discusses the application of flux-vector splitting for the Euler equations, a hyperbolic system of conservation laws. The author outlines two primary models for upwind differencing: the Riemann approach, which uses discrete waves to model interactions between computational cells, and the Boltzmann approach, which models interactions through the mixing of pseudo-particles. The focus is on the Boltzmann approach, specifically the flux-vector splitting technique, which distinguishes between forward and backward-moving particles. The goal is to split the flux \( f(w) \) into forward flux \( f^+ \) and backward flux \( f^- \) such that: 1. \( f(w) = f^+ + f^- \) 2. \( df^+ / dw \) has all eigenvalues \(\geq 0\), \( df^- / dw \) has all eigenvalues \(\leq 0\) 3. \( f^{\pm}(w) \) must be continuous and mimic the symmetry of \( f \) with respect to Mach numbers \( M \) 4. \( df^{\pm} / dw \) must be continuous 5. \( df^{\pm} / dw \) must have one eigenvalue vanish for \( |M| < 1 \) 6. \( f^{\pm}(M) \) must be a polynomial in \( M \) of the lowest possible degree These restrictions ensure that the splitting is unique and leads to smooth numerical results, particularly in supersonic regions. The paper also provides a derivation of the full, forward, and backward fluxes for the one-dimensional Euler equations.The paper by Bram van Leer discusses the application of flux-vector splitting for the Euler equations, a hyperbolic system of conservation laws. The author outlines two primary models for upwind differencing: the Riemann approach, which uses discrete waves to model interactions between computational cells, and the Boltzmann approach, which models interactions through the mixing of pseudo-particles. The focus is on the Boltzmann approach, specifically the flux-vector splitting technique, which distinguishes between forward and backward-moving particles. The goal is to split the flux \( f(w) \) into forward flux \( f^+ \) and backward flux \( f^- \) such that: 1. \( f(w) = f^+ + f^- \) 2. \( df^+ / dw \) has all eigenvalues \(\geq 0\), \( df^- / dw \) has all eigenvalues \(\leq 0\) 3. \( f^{\pm}(w) \) must be continuous and mimic the symmetry of \( f \) with respect to Mach numbers \( M \) 4. \( df^{\pm} / dw \) must be continuous 5. \( df^{\pm} / dw \) must have one eigenvalue vanish for \( |M| < 1 \) 6. \( f^{\pm}(M) \) must be a polynomial in \( M \) of the lowest possible degree These restrictions ensure that the splitting is unique and leads to smooth numerical results, particularly in supersonic regions. The paper also provides a derivation of the full, forward, and backward fluxes for the one-dimensional Euler equations.