Density distribution across the entire domain for PIROCK with different tolerances, the explicit approach, and reference solutions (top left). The top-middle panel depicts the distribution of the logarithm of the absolute difference, log₁₀(|ρ − ρ^ref|), between the different scenarios and the reference solution. The top-right panel shows the distribution of the logarithm of the absolute difference between analytical and reference solution log₁₀(|ρ − ρ^ref|). The bottom panels mirror the top ones but focus specifically on the shock region between x=0.82 and 0.88 (indicated by the two vertical blue lines in the top-left panel). The red dashed line represents the reference solution, and the black dashed line corresponds to the explicit. The remaining colors represent different PIROCK cases with varying tolerances.