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Contact us:
J.D. Maldonado
F.X.Timmes, my vitae
Approximately 60 years ago, von Neumann (1941), Taylor (1941) and Sedov (1946) independently derived a self-similar description of the evolution of the blast wave arising from a powerful explosion in a cold, uniform density background (also see Bethe et al 1947). They treated the explosion as an instantaneous release of energy at a point and assumed that the background material through which the expanding blast sweeps behaves as an ideal polytropic fluid. It is remarkable that this model yields exact, although algebraically complicated, analytical expressions for the fluid quantities.
Here we (Kamm, Bolstad, & Timmes, submitted ApJ Supplement, May 2007) describe the generation of robust numerical solutions for a Sedov blast wave propagating through a density gradient ρ=ρ0r-ω in planar, cylindrical or spherical geometry for the standard, singular and vacuum cases. In the standard, case a nonzero solution extends from the shock to the origin, where the pressure is finite. In the singular case, a nonzero solution extends from the shock to the origin, where the pressure vanishes. In the vacuum case a nonzero solution extends from the shock to a boundary point, where the density vanishes.
Four Sedov functions describe the spatial variation of density, material speed, and pressure with distance at any point in time. Although the Sedov functions are analytic, they appear to become singular for various combinations of parameters and at lower limits of integration. All six of these singularities are removable; they are a consequence of the way the solution is formulated. Expressions and techniques for the removal of the apparent singularities are described. In addition, numerical calculations of Sedov solutions require extended precision arithmetic near the origin to avoid running out of significant figures.
The venerable Sedov problem might appear to be an old solved problem. However, there is a paucity of papers that discuss all possible families of real solutions, in all common geometries, and address all the removable singularities. The constant density, spherically symmetric Sedov blast wave stalwart test case in the verification of hydrodynamic codes. However, it is not a particularily difficult test for a modern hydrocode. In this paper we identify several new verification problems involving more challenging Sedov blast waves. To encourage their adoption within standard test suites we offer a modest software library to generate numerically robust and efficient Sedov solutions.
This Sedov code is released under LA-CC-07-020. You should read Jim Kamm's opus and David Book's irreverant paper on generating Sedov solutions.
Solutions as a function of scaled distance:
 planar case |
 cylindrical case |
 spherical case1 |
 spherical case1 |
 spherical case2 |
 spherical case3 |
 spherical case4 |
 Spherical case5 |
 entropies |
 alpha is not one! |
sedov3.f solves for a blast
wave in a variable medium. |
Standard, singular and vacuum solutions:
 standard cases |
 singular cases |
 vacuum cases |
Code verificaton efforts:
 shock down a density gradient |
 analytical and numerical |
 convergence study |
 1D evolution movie |
sedov3.f solves for a blast
wave in a variable medium. |
Density uniform mesh:
 density 240x240 cells |
 density 480x480 cells |
 density 960x960 cells |
 density error 240x240 cells |
 density error 480x480 cells |
 density error 960x960 cells |
 density asymmetry 240x240 |
 density asymmetry 480x480 |
 density asymmetry 960x960 |
Density adaptive mesh:
 density 240x240 cells |
 density 480x480 cells |
 density 960x960 cells |
 density error 240x240 cells |
 density error 480x480 cells |
 density error 960x960 cells |
 density asymmetry 240x240 |
 density asymmetry 480x480 |
 density asymmetry 960x960 |