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

This is the authors' abstract. We don't add key points for this paper.

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