Self-similar imploding solutions of the 1D compressible Euler equations with a far field cutoff

Abstract

Imploding solutions to the radially symmetric, isentropic, compressible Euler equations have been well-studied, inspired by the work of Guderley. However, these smooth imploding solutions are shown to be numerically unstable and difficult to compute in practice. On the other hand, the imploding solution of Kidder has a closed form solution and is numerically computable. But, it is unbounded in the far field. We consider Kidder's formulation in one dimension in which the unbounded far field condition is replaced with a constant density cutoff of the initial data. Strikingly, a non-centered rarefaction emerges from the cutoff and suppresses the implosion. We present an exact analytic solution to the problem with the cutoff and support our theoretical predictions with numerical simulations.