TESTS Module

Hand-derived closed form for Test 1's target quantity (a specific product of VSH inner products), evaluated over a colatitude grid at $\phi=0$ for comparison against TEST1_VSH and TEST1_GW.

Arguments

Return Value complex(kind=dp), DIMENSION(NTH)

Test 1's target quantity computed via the GWI-coefficient expansion in scalar spherical harmonics, rather than direct VSH DOT products -- must agree with TEST1_ANALYTIC and TEST1_VSH to machine precision. This is the check that specifically exercises GWI (see also TEST2_GW for the GWJ counterpart).

Arguments

Return Value complex(kind=dp), DIMENSION(NTH)

Test 1's target quantity computed directly from DOT products of VSH_TOR/VSH_POL_DN/VSH_POL_UP values -- must agree with TEST1_ANALYTIC to machine precision.

Arguments

Return Value complex(kind=dp), DIMENSION(NTH)

Hand-derived closed form for Test 2's target quantity (a different product of VSH inner products than Test 1, with nonzero $m$ content), evaluated over a colatitude grid at a fixed $\phi$ for comparison against TEST2_VSH and TEST2_GW.

Arguments

Return Value complex(kind=dp), DIMENSION(NTH)

Test 2's target quantity computed via the GWJ-coefficient expansion (combined with a GWI term) in scalar spherical harmonics -- must agree with TEST2_ANALYTIC and TEST2_VSH to machine precision. This is the check that specifically exercises GWJ.

Arguments

Return Value complex(kind=dp), DIMENSION(NTH)

Test 2's target quantity computed directly from DOT products of PVSH_POL/PVSH_TOR/PVSH_RAD values -- must agree with TEST2_ANALYTIC to machine precision.

Arguments

Return Value complex(kind=dp), DIMENSION(NTH)

Batch-consistency check: ASSOC_LEGENDRE_ALL vs. looped ASSOC_LEGENDRE calls, for $0\le\ell\le\ell_{max},\,0\le m\le\ell$, swept over NX points $x\in[-0.9,0.9]$. Writes every (l,m,x) triple's batch value, single-mode value, and absolute difference to OUTUNIT; fails (STATUS=1) if any relative error exceeds $10^{-12}$.

Arguments

Batch-consistency check: DDX_ASSOC_LEGENDRE_ALL vs. looped DDX_ASSOC_LEGENDRE calls, for $0\le\ell\le\ell_{max},\,0\le m\le\ell$, swept over NX points $x\in[-0.9,0.9]$. Same tolerance/output convention as BATCH_ALM_CONS.

Arguments

Batch-consistency check: GRAD_SSH_ALL vs. looped GRAD_SSH calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Batch-consistency check: L_SSH_ALL vs. looped L_SSH calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Batch-consistency check: PVSH_POL_ALL vs. looped PVSH_POL calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Batch-consistency check: PVSH_RAD_ALL vs. looped PVSH_RAD calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Batch-consistency check: PVSH_TOR_ALL vs. looped PVSH_TOR calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$ (both routines agree at $\ell=0$ too -- PVSH_TOR explicitly branches to return zero there rather than evaluating a 0/0 form). Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Batch-consistency check: SSH_ALL vs. looped SSH calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any $|Y_{batch}-Y_{single}|$ exceeds $10^{-13}$.

Arguments

Batch-consistency check: VSH_POL_DN_ALL vs. looped VSH_POL_DN calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Batch-consistency check: VSH_POL_UP_ALL vs. looped VSH_POL_UP calls, for $0\le\ell\le\ell_{max},\,-\ell\le m\le\ell$, swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Batch-consistency check: VSH_TOR_ALL vs. looped VSH_TOR calls, for $1\le\ell\le\ell_{max},\,-\ell\le m\le\ell$ (starts at $\ell=1$; VSH_TOR is identical to PVSH_TOR, already covered at $\ell=0$ by BATCH_PVSH_TOR_CONS), swept over NTH colatitude points at fixed $\phi=\pi/4$. Fails (STATUS=1) if any component's absolute difference exceeds $10^{-12}$.

Arguments

Verifies pointwise orthogonality of the poloidal and toroidal polar VSH members, $\mathbf{Y}_{\ell m}^{(+1)}\cdot\mathbf{Y}_{\ell m}^{(0)} = 0$ for every $\ell\ge1,\,m,\,\theta,\,\phi$ -- an exact analytic identity (poloidal is a pure gradient, toroidal a pure $\hat r\times$gradient, and $\nabla_\perp f\cdot\hat r\times \nabla_\perp f=0$ always), so any nonzero DOT value here is purely numerical error.

Arguments

Verifies scalar spherical harmonic orthonormality, $\int Y_{\ell_1}^m(Y_{\ell_2}^m)^{*}\,d\Omega = \delta_{\ell_1\ell_2}$, for every $0\le m\le\ell_1\le\ell_2\le\ell_{max}$. The $\phi$ integral is done analytically ($2\pi$ for matching $m$, by construction 0 otherwise, so only same-$m$ pairs are checked); the $\theta$ integral (in $u=\cos\theta$) uses a plain NQUAD-point midpoint rule via ASSOC_LEGENDRE_ALL, not SHGLQ -- sufficient here since only a scalar pass/fail diagonal check is needed, not a high-order transform.

Arguments

Verifies the orthogonal-rotation identity linking the standard and polar VSH bases can be inverted exactly: $$\sqrt{\tfrac{\ell}{2\ell+1}}\,\mathbf{Y}_{\ell m}^{\ell+1} + \sqrt{\tfrac{\ell+1}{2\ell+1}}\,\mathbf{Y}_{\ell m}^{\ell-1} = \mathbf{Y}_{\ell m}^{(+1)},$$ $$-\sqrt{\tfrac{\ell+1}{2\ell+1}}\,\mathbf{Y}_{\ell m}^{\ell+1} + \sqrt{\tfrac{\ell}{2\ell+1}}\,\mathbf{Y}_{\ell m}^{\ell-1} = \mathbf{Y}_{\ell m}^{(-1)},$$ i.e. rotating VSH_POL_UP/VSH_POL_DN back by the same angle used to build them from PVSH_POL/PVSH_RAD recovers the originals exactly.

Arguments