Runs the standard Sod problem, a grid-refinement test, conservation, the Sedov constants and the blast slope.

The 1D Euler equations d_t U + d_x F(U) = 0 for mass, momentum and energy (ideal gas, p = (gamma-1)(E - rho u^2/2)) are solved by a finite-volume HLLC / MUSCL / SSP-RK2 scheme, the same core as the 3D page. A diaphragm between two gas states (Riemann problem) produces a left rarefaction fan, a contact discontinuity and a right shock. The exact solution (Toro) comes from solving f(p*) = f_L + f_R + (u_R - u_L) = 0 for the star pressure; the dashed curves are that exact answer.

The Sedov-Taylor point blast releases energy E in gas of density rho0: dimensional analysis gives R(t) = xi0 (E t^2 / rho0)^(1/5), with xi0 = 1.033 for gamma = 1.4. The profiles are self-similar in r/R; the solver curve is compared with the similarity ODE solution. Simplifications: ideal gas, no radiation or viscosity, small numerical diffusion smears shocks over 2-3 cells, and near-vacuum at the blast centre uses a small mass floor. This is gas dynamics for astrophysics and engineering (supernova remnants, shock tubes), not a design tool.