Finite Volume Methods III: Flow and Discontinuities

This chapter finite volume methods (FVM) applications and theory for compressible flows with discontinuities. It clearly explains how governing equations are integrated over control volumes and how interfacial fluxes are computed using Riemann solvers. The Godunov method is introduced as an exact approach for resolving local Riemann problems, while its limitations are addressed through the Roe scheme, which employs linearized Jacobians. Advanced reconstruction techniques, such as MUSCL and WENO, are incorporated to improve spatial accuracy, and slope limiters are utilized to maintain numerical stability in the vicinity of shocks. Entropy-related deficiencies inherent to the Roe solver are resolved through flux correction strategies designed to suppress non-physical expansion shocks. The chapter further evaluates alternative flux schemes, including HLL, and HLLE, with emphasis on their relative robustness and dissipation properties. The chapter presented cases of supersonic flow over wedge and cylinder and flow inside nozzle.