Hyperbolic conservation laws with space-dependent fluxes: II. General study of numerical fluxes

Following the previous paper, this one continues to study numerical approximations to the space-dependent flux functions in hyperbolic conservation laws. The investigation is based on the wave propagation behavior, Riemann problem, steady flows, hyperbolic properties, cell entropy inequalities, along with such well known numerical fluxes as the Godunov, Local Lax–Friedrichs and Engquist–Osher. All these give rise to correct description for the consistency and monotonicity of numerical fluxes, which ensure properly confined numerical solutions. Numerical examples show that the accordingly designed fluxes resolve discontinuities and smooth solutions very precisely.