Programs

Driver for the BENCHMARK suite (opt-in VSH_BUILD_BENCHMARK CMake target, not part of ctest): runs every BENCH_* routine across a fixed sweep of $\ell_{max}$ values, writing one row per $\ell_{max}$ to ./benchmark/bench_*.dat (consumed by py/plot_benchmark.py for the manuscript's performance figure).

Driver for the convergence-vs-degree accuracy sweep (opt-in VSH_BUILD_CONVERGENCE CMake target, not part of ctest): calls the same TESTS batch-consistency routines the core ctest gate uses (BATCH_ALM_CONS, BATCH_SSH_CONS, BATCH_VSH_TOR_CONS, BATCH_VSH_POL_UP_CONS, BATCH_VSH_POL_DN_CONS), but at $\ell_{max}=300$ instead of the small, fixed values main.f90 uses to gate correctness. Each routine already loops internally over every degree $0\le\ell\le\ell_{max}$ in one call, so this produces dense per-degree error data up to 300 without needing an outer sweep here.

Worked example (2c): idealized superconducting-core flux expulsion. Ties to Elfritz (2016), cited in the manuscript, on field expulsion from superconducting neutron-star cores. Simplified to the standard "perfectly conducting sphere in a uniform ambient field" textbook problem: a sphere of radius $R$ totally excludes flux (idealized Meissner/Type-I limit -- real neutron-star cores are believed Type-II, with flux confined to quantized fluxoids rather than fully excluded; that fuller picture is not computed here). The boundary condition $B_r(R)=0$ for all $\theta$ (no field may cross into a perfect diamagnet) fixes the induced dipole moment in closed form: $\mu_{ind} = -B_{0,amb}R^3/2$, which superposes on the uniform ambient field to give the exterior field. Interior field is exactly zero. Purely magnetostatic: no induction equation, no time evolution.

Worked example (2a): synthesize the classic exterior vacuum magnetic dipole ($\ell=1,\,m=0$) from FORTVSH's poloidal basis functions and confirm (i) it reproduces the textbook closed form $B_r=2\mu\cos\theta/r^3,\,B_\theta=\mu\sin\theta/r^3$, and (ii) it is numerically pure -- its overlap with every other $\ell$ (at $m=0$) is zero to quadrature precision. No induction equation or time evolution is involved: this is a static field-synthesis and orthogonal-projection check only.

Worked example (2b): uniformly magnetized sphere $\leftrightarrow$ exterior dipole. Classic textbook magnetostatic boundary-value problem (e.g. Griffiths Ex. 6.1 / Jackson): a sphere of radius $R$ with uniform magnetization has a genuinely uniform interior field and a pure exterior dipole field. In the normalized units used here ($\mu_0/4\pi=1$, i.e. $B_{r,ext}=2\mu\cos\theta/r^3$), the magnetic-scalar-potential solution gives the closed-form relation $B_0\,(\text{interior uniform field}) = 2\mu/R^3$. Although there is no free surface current in this relation's derivation but the interior/exterior fields are not continuous in $B_\theta$ (physically: continuity is broken by the sphere's own bound magnetization surface current, $\mathbf{K}=\mathbf{M}\times\hat n$) -- $B_r$ is continuous at $r=R$, but $B_\theta$ has a jump. Both are checked below using FORTVSH's PVSH_RAD/PVSH_POL evaluated with the interior ($S_{in}=Ar^2$, uniform field) and exterior ($S_{out}=C/r$, dipole) stream functions. Purely static/magnetostatic: no induction equation, no time evolution.

Worked example (2e): Geppert-Wiebicke (GWI/GWJ) coefficients swept across a range of mode-number combinations, each checked against an independent formula (not just the closed-form GWI/GWJ code itself evaluated a second way).

Driver for the full TESTS validation suite (run via ctest, target vsh_validation): runs every cross-validation, batch-consistency, and identity check in TESTS, writes each one's full per-point data to ./validation/*.dat, prints a PASS/FAIL line per check, and exits non-zero if any check failed (the signal ctest uses for a pass/fail build gate).

Driver for mapping ASSOC_LEGENDRE_NORM_ALL's numerical stability across the $(\ell, m, \theta)$ parameter space (opt-in VSH_BUILD_STABILITY CMake target, not part of ctest). See STABILITY_FINDINGS.md at the repo root for the full writeup -- this exists because the documented "stable to $\ell\sim2700$" claim for ASSOC_LEGENDRE_NORM_ALL turned out to depend on $\theta$ as well as $\ell$: for $\theta$ close to the poles and $m$ in a band tied to the classical turning point $\sin\theta \approx m/\ell$ of the associated Legendre ODE, the forward-column recurrence can blow up by 100+ orders of magnitude even at moderate $\ell$.

Driver for mapping the UNNORMALIZED Bonnet recurrence's (ASSOC_LEGENDRE/ASSOC_LEGENDRE_ALL, and everything single-mode built on it) usable accuracy range across $(\ell, m, \theta)$ (opt-in VSH_BUILD_STABILITY CMake target, same as STABILITY_MAIN -- not part of ctest). See STABILITY_FINDINGS.md at the repo root.

Worked example (2d): arbitrary-field VSH spectral decomposition, reconstruction, and convergence.

Worked example (2f): pure-toroidal VSH spectral decomposition, reconstruction, and convergence -- companion to vsh_decomposition.f90 (example 2d), which decomposes onto the radial family. This example decomposes onto the toroidal family only (PVSH_TOR, coefficients a_tor) instead.