Red line is the solution given by Godunov method. Blue line is the exact Riemann problem solution. All discontinuities locate at x=0. W_L stands for initial primitive quantities (\rho,u,p) on the left side, W_R stands for initial primitive quantities (\rho,u,p) on the right side.

For each example, pictures on the left side are \rho vs x plot, pictures on the right side are u vs x plot

Example: Shock tube

Example: Two strong rarefaction

Example: Left rarefaction and right contact and shock wave

Example: Right half of Woodward and Colella problem

Example: Two shock case

Other examples found in Toro's book. Below are density plots.