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Both modes integrate the 1-D Euler equations of an ideal gas with a finite-volume method: MUSCL reconstruction (van Leer limiter), an HLLC approximate Riemann solver and a two-stage SSP Runge-Kutta step (CFL 0.4). The spherical case adds the geometric area factors and the pressure source term 2p/r.
Sedov-Taylor blast. For an intense point explosion in a uniform gas, dimensional analysis (Taylor 1941, Sedov 1946) leaves only R = xi0 (E t^2 / rho0)^(1/5). The profiles follow from three coupled ODEs in the similarity variable x = r/R; here they are integrated numerically from the Rankine-Hugoniot jump at x = 1, and xi0 is fixed by requiring the total energy to equal E. The solver is started from a small hot sphere and must reproduce that law without being told it.
Sod shock tube. A diaphragm separates two gas states (Sod 1978). When it bursts a shock, a contact surface and a rarefaction fan appear. The exact solution comes from solving one nonlinear equation for the pressure between the two nonlinear waves (Toro 2009, ch. 4).
Approximations, honestly. Sedov-Taylor needs a strong shock, so the model is cut off where the front pressure falls to ten times the ambient pressure; beyond that the real wave becomes an acoustic N-wave. Real explosions also involve ground reflection and winds, radiation, real-gas chemistry, and a source of finite size and mass; none of that is modelled. The ambient pressure used for the cutoff scales with density as for air at constant temperature. The numerical start uses a cold ambient gas and a small floor on density at the nearly empty centre (the mass added there is reported in the checks). Energy is quoted per spherical release in free air. Natural examples (meteoroid airbursts, volcanic blasts, fireworks) are shown only as physics: the energies are order-of-magnitude values.
References: G. I. Taylor, Proc. R. Soc. A 201 (1950); L. I. Sedov, Similarity and Dimensional Methods in Mechanics; G. A. Sod, J. Comput. Phys. 27 (1978); E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics (2009).