Closing the gap in the solutions of the strong explosion problem: An expansion of the family of second-type self-similar solutions

Shock waves driven by the release of energy at the center of a cold ideal gas sphere of initial density rho\propto r^{-omega} approach a self-similar (SLS) behavior, with velocity \dot{R}\propto R^delta, as R->\infty. For omega>3 the solutions are of the second-type, i.e., delta is determined by the requirement that the flow should include a sonic point. No solution satisfying this requirement exists, however, in the 3\leq omega\leq omega_{g}(gamma) ``gap'' (\omega_{g}=3.26 for adiabatic index gamma=5/3). We argue that second-type solutions should not be required in general to include a sonic point. Rather, it is sufficient to require the existence of a characteristic line r_c(t), such that the energy in the region r_c(t)\infty, and an asymptotic solution given by the SLS solution at r_c(t)<r<R and deviating from it at r<r_c may be constructed. The two requirements coincide for omega>omega_g and the latter identifies delta=0 solutions as the asymptotic solutions for 3\leq omega\leq omega_{g} (as suggested by Gruzinov03). In these solutions, r_c is a C_0 characteristic. It is difficult to check, using numerical solutions of the hydrodynamic equations, whether the flow indeed approaches a delta=0 SLS behavior as R->\infty, due to the slow convergence to SLS for omega~3. We show that in this case the flow may be described by a modified SLS solution, d\ln\dot{R}/d\ln R=delta with slowly varying delta(R), eta\equiv d delta/d\ln R<<1, and spatial profiles given by a sum of the SLS solution corresponding to the instantaneous value of delta and a SLS correction linear in eta. The modified SLS solutions provide an excellent approximation to numerical solutions obtained for omega~3 at large R, with delta->0 (and eta\neq0) for 3\leq omega\leq omega_{g}. (abridged)Comment: 10 pages, 11 figures, somewhat revised, version accepted to Ap

Stability properties of the ENO method

We review the currently available stability properties of the ENO reconstruction procedure, such as its monotonicity and non-oscillatory properties, the sign property, upper bounds on cell interface jumps and a total variation-type bound. We also outline how these properties can be applied to derive stability and convergence of high-order accurate schemes for conservation laws.Comment: To appear in Handbook of Numerical Methods for Hyperbolic Problem