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
4.3. Magnetohydrodynamics 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
About the Author: Keiichi Kitamura is an Associate Professor at Yokohama National University. His work focuses on developing numerical methods in computational fluid dynamics for high-speed, low-speed, and multiphase/supercritical/MHD flows, and he has proposed numerical flux functions, e.g., SLAU2 and Post Limiter.
He received his doctor of engineering from Nagoya University in 2008. After serving as a postdoctoral researcher at JAXA and Glenn Research Center, NASA, he was appointed an Assistant Professor at Nagoya University in 2012. Since 2014, he has served in his current position. He was honored with young researcher awards by the Japan Society for Aeronautical and Space Sciences in 2012, by the Society for Promotion of Space Science in 2018, and by the Japan Society of Fluid Mechanics in 2018. He was also awarded the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology (The Young Scientist's Prize) in Japan, and the Frontier Commendation from Fluids Engineering Division, The Japan Society of Mechanical Engineers, both in 2019.
