Advancement of Shock Capturing Computational Fluid Dynamics Methods: Numerical Flux Functions in Finite Volume Method

1. Introduction: Brief Review of Finite Volume Method in Computational Fluid Dynamics

2. Role and History of Numerical Flux Functions

2.1. Issue 1: Anomalous Solutions of Captured Shock and Heating at Hypersonic Speeds

2.2. Issue 2: Stiffness Problem at Low Speeds

3. Numerical Flux Functions for Ideal Gas

3.1. Godunov's Exact Riemann Solver

3.2. Central-difference formulas, and Lax-Friedrichs method

3.3. Flux Difference Splitting (FDS): Roe's Approximate Riemann Solver (and Entropy Fix) and Osher's Approximate Riemann Solver

3.4. Flux Vector Splitting (FVS): Steger-Warming, Van Leer, Hänel, Liou-Steffen (Original AUSM), Zha-Bilgen, and Toro-Vazquez methods

3.5. Harten-Lax-van Leer Family: HLL, HLLE, HLLEM, HLLC, HLLD, and HLLI

3.6. FDS/FVS Hybrid Advection-Upstream-Splitting-Method Family: AUSMDV, AUSM+, SHUS, LDFSS, AUSMPW+, AUSM+-up, SLAU, SD-SLAU, SLAU2, and HR-SLAU2

3.7. Others: Rotated Roe-HLL and Genuinely Multidimensional Splitting

4. Numerical Flux Functions Extended to Other Fluids

4.1. Multiphase Flows

4.2. Supercritical Fluids

5. Reconstruction and Slope Limiters

5.1. Monotone Upstream-centered Schemes for Conservation Laws, (Weighted) Least-Squares, Green-Gauss (G-G), and Green-Gauss/Least-Square methods

5.2. Conventional Limiters

5.3. Post Limiters