The paper by E.F. Toro, M. Spruce, and W. Speares addresses the restoration of the contact surface in the Harten, Lax, and van Leer (HLL) Riemann solver, a key component in computational fluid dynamics (CFD) for solving shock-capturing problems. The authors propose an improved version of the HLL-Riemann solver, referred to as the HLLC-Riemann solver, which retains the accuracy and robustness of the exact Riemann solver while being simpler and more computationally efficient, particularly for non-ideal gases. The original HLL-Riemann solver assumes a wave configuration consisting of two waves separating three constant states, neglecting intermediate waves. This approach is efficient for hyperbolic systems like the one-dimensional shallow water equations but fails to accurately capture contact surfaces in more complex systems such as the Euler equations. The HLLC-Riemann solver restores the contact surface by incorporating the intermediate wave, ensuring accurate resolution of discontinuities in temperature and internal energy, which is crucial for applications like combustion problems. The paper also presents efficient methods for estimating wave speeds and implements the HLLC-Riemann solver in the WAF method, a second-order TVD method of the Godunov type. Numerical results for one- and two-dimensional gas dynamics problems, including both ideal and covolume gases, are presented to validate the improved solver's performance. The authors conclude that the HLLC-Riemann solver is a robust and efficient tool for solving real gases in CFD simulations.The paper by E.F. Toro, M. Spruce, and W. Speares addresses the restoration of the contact surface in the Harten, Lax, and van Leer (HLL) Riemann solver, a key component in computational fluid dynamics (CFD) for solving shock-capturing problems. The authors propose an improved version of the HLL-Riemann solver, referred to as the HLLC-Riemann solver, which retains the accuracy and robustness of the exact Riemann solver while being simpler and more computationally efficient, particularly for non-ideal gases. The original HLL-Riemann solver assumes a wave configuration consisting of two waves separating three constant states, neglecting intermediate waves. This approach is efficient for hyperbolic systems like the one-dimensional shallow water equations but fails to accurately capture contact surfaces in more complex systems such as the Euler equations. The HLLC-Riemann solver restores the contact surface by incorporating the intermediate wave, ensuring accurate resolution of discontinuities in temperature and internal energy, which is crucial for applications like combustion problems. The paper also presents efficient methods for estimating wave speeds and implements the HLLC-Riemann solver in the WAF method, a second-order TVD method of the Godunov type. Numerical results for one- and two-dimensional gas dynamics problems, including both ideal and covolume gases, are presented to validate the improved solver's performance. The authors conclude that the HLLC-Riemann solver is a robust and efficient tool for solving real gases in CFD simulations.