We present a class of self-similar solutions describing ultrahigh compression of a uniform-density target by spherically converging,
stacked shock waves. Extending the classical Guderley model, we derive a scaling law for the final density of the form ρr /ρ0 ~
P^{β(N−1)}, where N is the number of shocks, P the stage pressure ratio, and β a numerical exponent determined by the adiabatic
index γ. One-dimensional hydrodynamic simulations confirm the validity of this scaling across a broad parameter range.
Notably, the relation remains accurate even in the strongly nonlinear regime up to P∼ 70, well beyond the perturbative limit,
highlighting the robustness and practical relevance of the model. Owing to its volumetric geometry, this compression scheme
inherently avoids the Rayleigh-Taylor instability, which typically compromises shell-based implosions, and thereby establishes a
theoretical benchmark for instability-free compression in inertial confinement fusion.