Experimental Benchmark and Numerical Validation of the Compressible Navier-Stokes-Madelung System via Root-Free Laplace Inversion on Chebyshev Nodes

Abstract: We present an exhaustive physical, mathematical, and numerical validation of the unified compressible Navier-Stokes-Madelung (NSM) system against classical hydrodynamic benchmarks and high-precision laboratory experiments. The model couples classical compressible Navier-Stokes dissipation (shear viscosity μ and bulk viscosity ζ) with the quantum Madelung stress tensor, rigorously deduced from first principles via the polar transformation of the time-dependent non-linear Schrödinger equation and the Korteweg capillarity tensor identity. Linearization over a quiescent equilibrium state yields a fourth-order dissipative-dispersive quantum-acoustic wave equation. By applying the unilateral Laplace transform in time coupled with Chebyshev-Gauss-Lobatto (CGL) spectral collocation in space, the boundary-value problem is formally proven to reduce to a strictly proper rational matrix transfer function P(s)/Q(s) with deg(P) < deg(Q), ensuring strict causality and regular time evolution without spurious delta-singularities. To overcome Wilkinson's catastrophic conditioning barrier in high-order polynomial root-finding (κ(Q) ~ K! > 10^20), all analytical solutions and spectral residuals were computed using the deterministic Root-Free Laplace Inversion algorithm implemented in the open-source MATLAB toolbox LaplaceReason (MathWorks File Exchange ID: 185102, DOI: 10.5281/zenodo.23760958). The formulation is benchmarked across four fundamental physical problems: 1. Sod Shock Tube (Riemann Problem): Demonstrates asymptotic recovery of classical gas dynamics (β -> 0, L^2 < 10^-14) and physical dispersive shock wave (DSW) regularization without artificial viscosity. 2. Alsmeyer (1976) Shock Thickness in Argon (Mach 1.55 - 9.00): Resolves the historical failure of classical Navier-Stokes, which underpredicts shock thickness by over 284% at Mach 9; the Madelung dispersive term accurately reproduces experimental saturation (λ_1/δ_ρ ≈ 0.50) with relative error < 1.1%. 3. Sonoluminescence Collapse (Barber & Putterman, PRL 1991): Halts the unphysical Rayleigh-Plesset mathematical collapse singularity at finite radius R_min ≈ 0.55 μm (χ^2 = 0.53), explaining the conversion of acoustic compression into coherent picosecond photon bursts. 4. Spectral Convergence: Achieves exponential spectral convergence down to machine precision (ε ~ 10^-15) at N = 128 nodes, unconditionally stable and free of Courant-Friedrichs-Lewy (CFL) time-step limitations. Reproducibility & Open Science: This deposit contains the complete manuscript (PDF, DOCX, LaTeX), all four experimental datasets in CSV format, 300 DPI publication-quality figures, and Python/MATLAB validation scripts. Companion Software & Datasets: - MATLAB Toolbox (MathWorks): https://es.mathworks.com/matlabcentral/fileexchange/185102-laplacereason-continuous-dynamics-stiff-ode-benchmark - Interactive Cloud Execution (MATLAB Online): https://matlab.mathworks.com/open/fileexchange/v1?id=185102&file=demo_laplacereason_interactive.m - Software Archive DOI: 10.5281/zenodo.23760958 - Theoretical Foundation DOI: 10.5281/zenodo.23661735