New Analytic Solution for Compressible Euler Equations with Bounded Far Field
Researchers have derived an exact analytic solution showing that replacing the unbounded far-field condition in Kidder's imploding solution with a constant-density cutoff causes a non-centered rarefaction wave to emerge and suppress the implosion. Kidder's solution, unlike classical Guderley-type imploding solutions, has a closed form and is numerically tractable, but its unbounded far-field behavior has been a practical obstacle. The finding matters because it reveals a fundamental sensitivity of implosion dynamics to far-field boundary conditions, with implications for numerical simulation and physical modeling of converging shocks.
A new preprint posted to arXiv examines self-similar imploding solutions of the one-dimensional radially symmetric, isentropic, compressible Euler equations — a class of problems with roots in Guderley's foundational work and relevance to inertial confinement fusion and shock physics. While Guderley-type smooth imploding solutions are known to be numerically unstable and difficult to compute, Kidder's formulation offers a closed-form alternative; however, it is unbounded in the far field, limiting its practical applicability. The authors modify Kidder's one-dimensional setup by replacing the unbounded far-field condition with a constant-density cutoff on the initial data. Strikingly, this seemingly minor modification causes a non-centered rarefaction wave to emerge from the cutoff location, which propagates inward and ultimately suppresses the implosion entirely. The paper presents a fully exact analytic solution to this modified problem and validates the theoretical predictions with numerical simulations across 25 pages and 8 figures. The result highlights how far-field boundary conditions can qualitatively alter implosion dynamics, a consideration of potential importance in both mathematical analysis and computational physics applications.
What's missing
The study is a preprint and has not yet undergone peer review. The work focuses on the one-dimensional case; whether analogous suppression behavior occurs in two or three spatial dimensions under similar cutoff conditions is not addressed.
What different sources said
- arXiv physicsCenter
Self-similar imploding solutions of the 1D compressible Euler equations with a far field cutoff
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