Numerical Schemes for Hydrodynamics Based on Multi-Dimensional Riemann Solvers

Here, U ≡ (v,ux,uy,e), superscript T stands for transpose, r is mass density, v ≡ 1/ρ, ux and uy are the components of flow velocity, e is the specific total energy, and Fx and Fy are fluxes in the xand ydirections respectively. A 2-D Riemann problem is Eq.(1) with a set of constant states for each region surrounding a point; for example, four constant states in the four quadrants in a structured mesh. At each time step within a simulation, what are known are mass, momentum, and energy in each numerical cell. The initial condition surrounding each grid point naturally corresponds to a 2-D Riemann problem.