Shock tube problem

Shock tube problem is a special case of Riemann problem with velocities on both sides of discontinuity set to zero. It is often used as a test case for validation of numerical codes, because analytical solutions are available.The initial condition is given by

$u=u_L=0 , p=p_L , \rho=\rho_L, x<0,$
$u=u_R=0 , p=p_R , \rho=\rho_R, x>0.$

A well-known special case is the Sod problem (Sod, 1978) with initial conditions

$p_L=1, \rho_L=1, p_R=0.1, \rho_R=0.125.$