Evidence
Validation
The numbers behind Conatus, in one place. Each value is compared with a published result or an independent code, and the deviation is the difference between them.
Linear stability
Eigenvalues and critical parameters of classical flows, against published tables.
| Test | Value obtained | Reference | Deviation |
|---|---|---|---|
Orr–Sommerfeld, Re = 10⁴, α = 1SourceVerification table of the Conatus paper; deviation: as stated in the row | c = 0.23752648882+0.00373967062i (8 digits) | Orszag 1971 | 8 digits |
Critical Reynolds number, plane PoiseuilleSourceVerification table of the Conatus paper | Rec = 5772.2218, αc = 1.020547 | 5772.22 | — |
Kolmogorov flow 2D, n = 4SourceVerification table of the Conatus paper | laminar threshold Relin = 9.9669 bracketed | Chandler & Kerswell 2013 | — |
Hagen–Poiseuille, (n,k) = (1,1),(1,0), R = 10²–10⁴SourceVerification table of the Conatus paper; deviation: as stated in the row | rightmost eigenvalues within 6.3×10⁻¹¹ (table rounding) | Meseguer & Trefethen 2003 Table 1 | 6.3×10⁻¹¹ |
Hagen–Poiseuille, R = 1000, 8 eigenvaluesSourceVerification table of the Conatus paper; deviation: as stated in the row | 9.4×10⁻¹⁴ from Dedalus | Dedalus | 9.4×10⁻¹⁴ |
Taylor–Couette, η = 0.5SourceVerification table of the Conatus paper | Rec = 68.1863, kc = 3.1625 (axisymmetric) | 68.19, 3.16 | — |
Rayleigh–Bénard onset (rigid)SourceVerification table of the Conatus paper | Rac = 1707.76178, kc = 3.116324; same in the z → −z subspace | 1707.76 Reid & Harris 1958 | — |
Rayleigh–Bénard onset, rigid bottom, stress-free topSourceVerification table of the Conatus paper | Rac = 1100.6496, kc = 2.6823 | 1100.65, 2.682 Chandrasekhar 1961 | — |
Stress-free Rayleigh–Bénard, kc = π/√2 (sine/cosine; Chebyshev + free_slip)SourceVerification table of the Conatus paper | growth rate vs closed form 4.6×10⁻¹⁵; 3×10⁻¹³; Dedalus 3×10⁻¹²; Rac = 27π⁴/4 bracketed | Chandrasekhar 1961 | — |
Rayleigh–Bénard 2D, Ra = 10⁵SourceVerification table of the Conatus paper; deviation: as stated in the row | Nu = 4.99432, agreement with Dedalus 10⁻⁸ | Dedalus | 10⁻⁸ |
Rotating convection onset, shell ri/ro = 0.35, E = 10⁻³, Pr = 1 Christensen et al. 2001SourceVerification table of the Conatus paper | mc = 4, Rac = 55.904299, ωc = −0.0230918 Ω (converged in N, L; m = 3,5: 57.957, 58.493) | 55.9 Skene & Tobias 2025; 55.904296 Barik et al. 2023 | — |
Same, equatorially symmetric subspaceSourceVerification table of the Conatus paper; deviation: as stated in the row | Rac = 55.904299 (1.2×10⁻¹³ from the full group), half the unknowns; antisymmetric class decays, σ = −35.3 | Barik et al. 2023; Busse 1970 | 1.2×10⁻¹³ |
3D channelSourceVerification table of the Conatus paper | Orszag 8 digits, Squire 7×10⁻¹¹, Dedalus 3D 7.6×10⁻¹⁴ | Dedalus | — |
Lane–Emden n = 3SourceVerification table of the Conatus paper; deviation: as stated in the row | ξ1 = 6.89684861937693 (5×10⁻¹⁵) | Horedt 2004 | 5×10⁻¹⁵ |
Time integration
Orders of the time steppers and agreement with Dedalus.
| Test | Value obtained | Reference | Deviation |
|---|---|---|---|
Heat equation, SBDF1–4SourceVerification table of the Conatus paper | observed orders 0.99, 2.07, 2.98, 3.96 (SBDF3/4 with the RK443 start-up; 1.98, 1.99 with the default lower-order start-up) | theory | — |
Heat equation, CNAB1, CNAB2, MCNAB2, CNLF2, RKSMR, RKGFYSourceVerification table of the Conatus paper; deviation: as stated in the row | 1.07, 2.00, 2.00, 2.01, 2.01, 2.00; Dedalus to ≤ 1.1×10⁻¹⁴ | theory, Dedalus | 1.1×10⁻¹⁴ |
Heat equation, both boundary values movingSourceVerification table of the Conatus paper | RK443 2.76, 2.81 (2.0 with g at the stage times, 1.05 in the Dedalus convention); RK222, RKSMR, RKGFY 2.00; disk 2.98, shell 2.85 | theory | — |
Geometries and bases
The disk, annulus, sphere, shell and ball, the whole and half line, compound intervals and stress-free walls, against closed forms and Dedalus.
| Test | Value obtained | Reference | Deviation |
|---|---|---|---|
Harmonic oscillator on the line (Hermite)SourceVerification table of the Conatus paper; deviation: as stated in the row | En = 2n+1, 20 values to 2.8×10⁻¹⁴ | exact | 2.8×10⁻¹⁴ |
Heat kernels on the line and the half line (Hermite, Laguerre)SourceVerification table of the Conatus paper | 1.7×10⁻⁹ (N = 48); half line < 10⁻⁸; Laguerre BVP e−x exact | exact | — |
Interior layer tanh(20x), compound intervalSourceVerification table of the Conatus paper | 2.5×10⁻¹⁴ on 3 pieces vs 1.3×10⁻⁷ on one interval (N = 192) | exact | — |
Stress-free curved walls (disk, annulus, pipe, ball, shell)SourceVerification table of the Conatus paper | solid-body rotation steady to ≤ 8×10⁻¹⁴; Stokes rates ≤ 1.1×10⁻¹⁰; ball/shell vs Dedalus 3.2×10⁻¹³/3.3×10⁻¹³ | Bessel, Dedalus | — |
Disk, Zernike basisSourceVerification table of the Conatus paper | −Δ eigenvalues jm,n² at 2.5×10⁻¹³ for m = −3…12; Poisson MMS 1.3×10⁻¹⁴ | Abramowitz & Stegun 1964 9.5 | — |
AnnulusSourceVerification table of the Conatus paper | Dirichlet eigenvalues at 8×10⁻¹⁵; Poisson MMS 6×10⁻¹² | Abramowitz & Stegun 1964 9.5.27 | — |
Nonlinear diffusion on the diskSourceVerification table of the Conatus paper; deviation: as stated in the row | max|u−uDedalus| = 1.7×10⁻¹⁴ after 50 RK443 steps | Dedalus | 1.7×10⁻¹⁴ |
Navier–Stokes 2D on the diskSourceVerification table of the Conatus paper | 1.3×10⁻¹³ from Dedalus, |∇·u| ≤ 4×10⁻¹³ | Dedalus | — |
Sphere, spin-weighted harmonicsSourceVerification table of the Conatus paper; deviation: as stated in the row | −Δ: ℓ(ℓ+1)/R² and (ℓ(ℓ+1)−1)/R² (vectors) to 10⁻¹² | exact | 10⁻¹² |
Ball, 3D ZernikeSourceVerification table of the Conatus paper; deviation: as stated in the row | Dirichlet −Δ: jℓ,n² to 10⁻¹¹; no-slip vector: zeros of jℓ−1,jℓ,jℓ+1 | Abramowitz & Stegun 1964 10.1 | 10⁻¹¹ |
ShellSourceVerification table of the Conatus paper | Dirichlet −Δ: cross-product zeros to 10⁻¹¹; Poisson MMS 10⁻¹² | Abramowitz & Stegun 1964 10.1 | — |
Nonlinear diffusion, sphere and ballSourceVerification table of the Conatus paper | 2×10⁻¹⁴ and 4×10⁻¹⁵ from Dedalus after 50 RK443 steps | Dedalus | — |
Navier–Stokes, ball and shellSourceVerification table of the Conatus paper | 1.3×10⁻¹⁴ and 2.2×10⁻¹³ from Dedalus | Dedalus | — |
Shallow water on the sphereSourceVerification table of the Conatus paper; deviation: as stated in the row | 7.2×10⁻¹⁵ from Dedalus after 50 RK443 steps | Dedalus | 7.2×10⁻¹⁵ |
Rossby–Haurwitz wave, R = 4 Williamson et al. 1992SourceVerification table of the Conatus paper | 14 days: mass exact, energy 1.4×10⁻⁷; pattern 7.7% slower than the non-divergent speed, converged (T42/T63 within 0.13%) | Haurwitz (approximate) | — |
Dynamics
Periodic orbits, bifurcations, Floquet exponents and exact adjoints, including the spherical shell.
| Test | Value obtained | Reference | Deviation |
|---|---|---|---|
Mathieu equationSourceVerification table of the Conatus paper; deviation: as stated in the row | Floquet exponents at 2–8×10⁻¹⁴ of the monodromy | Abramowitz & Stegun 1964 Table 20.1 | 8×10⁻¹⁴ |
Pulsatile scalar, M = 2 harmonicsSourceVerification table of the Conatus paper; deviation: as stated in the row | Hill vs monodromy < 10⁻⁸ | — | 10⁻⁸ |
Discrete adjointSourceVerification table of the Conatus paper | ⟨Jv,w⟩−⟨v,JTw⟩ ≤ 10⁻¹³, Taylor order 2.000; Burgers on Chebyshev, Legendre, Jacobi, compound, Hermite and Laguerre axes ≤ 9×10⁻¹⁴ (relative to |Jv||w|) | — | — |
Exact adjoint in the spherical shell: Dot product and TaylorSourceVerification table of the Conatus paper | at (Nϕ,Lmax,Nr) = (16,7,12), Ekman number E = 10⁻², Coriolis and u·∇u explicit, ten steps: ≤ 3.5×10⁻¹³ (RK443, SBDF2, admissible directions; 6.9×10⁻¹³ relative to the norms for random full states), Taylor order 1.9996 → 2.0000. Coriolis may also be implicit (left-hand side, one block per m): same results. With the m = 0 projection in the step, the dot-product test gives 2.5×10⁻¹² (RK443) and 1.3×10⁻¹³ (SBDF2) relative to the pairing (9×10⁻¹⁵ and 1.4×10⁻¹⁵ relative to the product of the norms), from random complex directions whose m = 0 coefficients are not real. With the planned right-hand side the test gives 4.5×10⁻¹² relative to the product of the norms, the rounding of the flow map itself. | — | 3.5×10⁻¹³ |
Exact adjoint in the spherical shell: Self-adjoint Stokes flowSourceVerification table of the Conatus paper | (E = 1, r∈[0.5,1.5], T = 0.1): λ1 = 12.071011094164 (toroidal ℓ = 1, j1(kri)y1(kro) = j1(kro)y1(kri)), confirmed as the least-damped mode by the eigenvalue solver. The discrete optimal gain exceeds e^{−2λ1T} by 1.2×10⁻⁵, 2.5×10⁻⁶, 7.6×10⁻⁷ at Nr = 16,24,32: algebraically (≈ N−4.0), while λ1 itself is exact to 5.5×10⁻⁹ at Nr = 16 and the adjoint is exact: the first-order tau discretisation is energy-self-adjoint only to O(N⁻²), and the gain sees its square. At Nr = 128, Δt = 10⁻⁴ (RK443, 1000 steps) the gain is 0.089438663224 against e^{−2λ1T} = 0.089438662948, 3.1×10⁻⁹ apart. A gain reported in a shell must therefore show its convergence in Nr. | — | 3.1×10⁻⁹ |
Exact adjoint in the spherical shell: RotationSourceVerification table of the Conatus paper | The energy adjoint of the rotating map is the map with −Ω: PW⁻¹J†Wv matches it to 7.1,4.4,2.1,1.2×10⁻³ (Nr = 12–32, smooth v), tracking the non-rotating mismatch (4.1×10⁻³–6.7×10⁻⁴), and is 0.97 away from the +Ω map. | — | — |
Exact adjoint in the spherical shell: Nonlinear optimal perturbationsSourceVerification table of the Conatus paper | Navier–Stokes, E = 1, T = 0.1: the gain found from a random seed equals the linear optimal gain (Lanczos on PW⁻¹J†WJ) to 2.2×10⁻¹¹, 7.9×10⁻¹³, 7.9×10⁻¹³ for E0 = 10⁻⁶,10⁻¹⁰,10⁻¹⁴ in 11–12 iterations. | — | 2.2×10⁻¹¹ |
Kuramoto–Sivashinsky L = 22SourceVerification table of the Conatus paper | E1–E3, TW±1,2 to 4 digits; shortest pre-periodic orbit, residual 4×10⁻¹⁴ | Cvitanović et al. 2010 | — |
KS shortest pre-periodic orbit, Floquet multipliersSourceVerification table of the Conatus paper | μ1 = 0.0332085, μ5 = −0.216373, μ6,7 = −0.265245, θ6,7 = ±2.6205: all published digits | Ding & Cvitanović 2016 Table 1 | — |
KS one-cell steady stateSourceVerification table of the Conatus paper | Fourier coefficients vs published values | Greene & Kim 1988 | — |
Bratu on [−1,1]SourceVerification table of the Conatus paper; deviation: as stated in the row | fold λc = 0.87845767978129 at 3×10⁻¹⁴ | closed form | 3×10⁻¹⁴ |
KS folds of E1 (points K, I)SourceVerification table of the Conatus paper; deviation: as stated in the row | α = 34.16914 (1.8×10⁻⁶), 36.23502 | Kevrekidis et al. 1990 | 1.8×10⁻⁶ |
KS pitchfork of E3SourceVerification table of the Conatus paper; deviation: as stated in the row | L = 18.84959 (6π at 4×10⁻⁵) | Cvitanović et al. 2010 | 4×10⁻⁵ |
Patterns and other checks
Pattern-forming systems, operator compilers and symmetry subspaces.
| Test | Value obtained | Reference | Deviation |
|---|---|---|---|
Symmetry subspaces (equatorial, m0-fold, axial)SourceVerification table of the Conatus paper; deviation: as stated in the row | reduced spectra of −Δ exact; Navier–Stokes (disk, ball, shell, pipe), heat, shallow water: reduced and full trajectories agree to ≤ 10⁻¹² | exact; full space | 10⁻¹² |
Exact rates, every way into M and L (24 cases)SourceVerification table of the Conatus paper | rates to 10⁻⁷–10⁻¹⁰ in all geometries and symmetry subspaces | closed forms | — |
Closed forms on every execution pathSourceVerification table of the Conatus paper | CPU, fused, CUDA and CUDA graphs (RK443, SBDF2); non-normal 2×2 gain G(t) to 10⁻⁸; Mathieu (DLMF 28.15.1 to q⁶), Meissner, Hill with M ≠ I to 10⁻¹²; adjoint gradients to 4×10⁻⁸; fold/Hopf continuation to 10⁻⁹ | closed forms | — |
IMEX coefficientsSourceVerification table of the Conatus paper; deviation: as stated in the row | RK111, RK222, RK443 equal to the printed tableaux (2 ulp), SBDF1–4 exact; the coefficients of all thirteen schemes equal those of Dedalus to 4.5×10⁻¹⁶; explicit part alone (ut = −u², RK443 start-up): SBDF1–4 1.01, 2.01, 3.00, 3.96, RK111/222/443 1.01, 2.03, 3.03 | Ascher et al. 1997; Wang & Ruuth 2008 | 4.5×10⁻¹⁶ |
Curvilinear transformsSourceVerification table of the Conatus paper | |FB−I|: sphere 5×10⁻¹⁵ (L = 64), disk 6×10⁻¹⁵ (N = 40), ball 3×10⁻¹⁵ (N = 32); refined Gauss nodes and long-double Jacobi normalisations (library weights and log-gamma norms gave 1.6×10⁻¹³, 1.1×10⁻¹³, 6×10⁻¹⁴) | rounding | — |
How these numbers are produced
Values are taken from the validation record of Conatus 0.1.1 (the Verification table of the Conatus paper) at every build of this site, never typed by hand. A deviation is the relative error of a recorded measurement or the one deviation the row states; a dash means the row states the agreement in words rather than as one number. Each number was produced by a recorded run; the run records (inputs, code version, log) are kept with the development repository, not in this one.