The Sod Gasdynamics Problem as a Tool for Benchmarking Face Flux Construction in the Finite Volume Method

The finite volume method approach in computational fluid dynamics to numerically model a fluid flow problem involves the process of formulating the numerical flux at the faces of the control volume. This process is important in deciding the resolution of the numerical solution, thus its quality. In the current work, the performance of different flux construction methods when solving the one-dimensional Euler equations for an inviscid flow is analyzed through a test problem in the literature having an exact (analytical) solution, which is the Sod problem. The considered flux methods are: exact Riemann solver (Godunov), Roe, Kurganov-Noelle-Petrova, Kurganov and Tadmor, Steger and Warming flux-vector splitting, van Leer flux-vector splitting, AUSM, AUSM+, AUSM+-up, AUFS, five variants of the Harten-Lax-van Leer (HLL) family, and corresponding five variants of the Harten-Lax-van Leer-Contact (HLLC) family, Lax-Friedrichs (Lax), and Rusanov. The methods of exact Riemann solver and van Leer showed excellent performance. The Riemann exact method took the longest runtime, but the spread of runtime among all methods was not large.