Benchmarks
VILMA2 is validated against the published GIA community benchmarks: the Charles University / Spada et al. (2011) intercomparison (Spada et al. 2011), the sea-level-equation intercomparison of Martinec et al. (2018) (Martinec et al. 2018) and, for lateral viscosity, the low-viscosity-zone case of Weerdesteijn et al. (2023) (Weerdesteijn et al. 2023). Validation proceeds along a ladder. Each rung is an independent benchmark that exercises one more piece of physics, so a failure localizes to the rung just added.
Every benchmark is an executable test with a pass/fail exit. The pages below quote two numbers for each: the tolerance the test asserts, and the difference measured with the reference build (model 1790f86, ifx 2024.1, -Ofast -march=znver2, on the AWI cluster albedo). The tolerance is the claim the test suite protects. The measured difference shows how much margin there is.
The reference datasets are vendored in the repository under data/benchmarks/ with provenance and licensing in data/benchmarks/PROVENANCE.md. None of it is VILMA2 output.
Summary results
| # | Benchmark | Physics exercised | Reference | Test (suite) | Asserted | Measured |
|---|---|---|---|---|---|---|
| 1 | SH transform round trip | spectral kernel | analytic | test_sht (check) |
\(10^{-12}\) | \(\sim 10^{-15}\) |
| 2 | Love numbers \(h,\ell,k\) | elastic and fluid radial solve, self-gravity | M3–L70–V01 table (Spada et al. 2011) | test_benchmark_love (check) |
1 %: elastic \(h,k\) at degrees 2–48, fluid \(h,\ell,k\) at 2–8 | \(h\) ≤ 0.08 % to degree 256; \(k\), \(\ell\) exceed 1 % above degrees 96, 144 |
| 3 | Disc-load response | spatial synthesis, time relaxation | Spada disc test (Spada et al. 2011) | test_benchmark_disc (check) |
elastic profile 6 % (\(u\)), 4 % (\(N\)); center uplift 3 % at 1–10 kyr | 4.0 %, 2.8 %; ≤ 0.6 % |
| 4a | Cap loading, case A | vertical and horizontal displacement, geoid | Martinec 2018 fig. 10 (Martinec et al. 2018) | test_benchmark_martinec (check) |
1.5 % (\(u\), \(v\)), 3 % (\(N\)) | 0.5 %, 0.7 %, 1.9 % |
| 4b | Sea-level equation | migrating coastline, flotation, sub-grid coast | Martinec 2018 cases C2, D3, E2, F1 (Martinec et al. 2018) | test_benchmark_sle (check-slow) |
30 % (\(u\)), 35 % (\(v\)), 12 % (\(N\)), 8 % (sea surface), 12 % (rsl) | D3, E2, F1: ≤ 7.6 %, rsl ≤ 1.8 %; C2: ≤ 28 %, rsl ≤ 7.2 % |
| 5 | Rotation / polar motion | Liouville equation, tidal Love numbers | Spada test 3/2, Table 14 (Spada et al. 2011) | test_rotation (check) |
3 % in \(\lvert m(t)\rvert\), 2 % in rigid inertia | ≤ 0.9 %, ≤ 0.4 % |
| 6 | Low-viscosity zone | laterally varying viscosity | ASPECT, Abaqus, TABOO (Weerdesteijn et al. 2023) | test_benchmark_lvz (check-slow) |
8 % in center displacement | 2.6 %, 1.0 % |
All percentages are relative to the peak of the reference along the compared profile, or to the reference value itself for single numbers. The per-benchmark pages define each measure and give the full tables.
What is validated and what is open
Validated. The radial solver, the Maxwell time stepping, the spatial synthesis, the horizontal displacement, the sea-level equation with a fixed and a migrating coastline, the Liouville polar-motion solve and the laterally varying viscosity each pass a published benchmark at the tolerances above. The two migrating-coastline cases and the deep-basin fixed-ocean case agree with the reference to below 1 m in every field.
Open.
- Case C2 of the sea-level benchmark. The model differs from the reference by a uniform 5.9 m in sea surface (4.9 % of the signal) and by a degree-1 pattern of about 7 m in uplift. The test passes only because its uplift and horizontal tolerances are 30 % and 35 %. The cause is not attributed (see Sea-level equation).
- Love numbers \(k\) and \(\ell\) at high degree. They leave 1 % of the normal-mode table above degrees 96 and 144 at the default radial mesh, while \(h\) stays within 0.1 %. The error is second order in the element size and falls below 1 % to degree 256 with a mesh four times finer (see Love numbers). The asserted range of the test (degree 48) has not been extended.
- The disc response at 100 kyr and the uplift rate are not compared (see Disc-load response).
- Three-dimensional viscosity. The only comparison with other codes is the axisymmetric low-viscosity-zone case on a 200 yr time scale, and a deglaciation comparison with VILMA1. No multi-code benchmark of a realistic 3-D structure at glacial time scales has been run.
- Toroidal flow. No community benchmark exercises it. It is checked by internal consistency, symmetry and scaling tests (see Toroidal flow) and by the VILMA1 comparison of the last deglaciation, and is on by default. Its effect there is up to 92 m of relative sea level locally, 2–3 m rms globally.
- Rotation coupled to the sea-level equation has no published benchmark and is checked for consistency only.
How to read the rungs
- Rungs 1–2 isolate the linear algebra and the radial physics: the transform, then the per-degree elastic and fluid Love numbers. The fluid limit of a homogeneous sphere is an analytic anchor (\(h_j\to-(2j+1)/3\), \(k_j\to-1\)).
- Rung 3 integrates the per-degree response over a spatial load (the disc) and adds the time relaxation.
- Rung 4 adds the horizontal displacement (case A) and then the full self-consistent sea-level equation: ocean function, mass conservation, coastline migration, flotation, and the sub-grid sloping coast.
- Rung 5 adds the rotational feedback. It is on by default for real-Earth runs and off in the non-rotating benchmarks.
- Rung 6 adds laterally varying viscosity.
Rungs 2, 3 and 5 compare with normal-mode solutions, which are accurate to their tabulated digits at low degree. Rungs 4b and 6 are comparisons between numerical codes. The reference is one code’s output, and the published scatter between codes (a few percent) bounds what agreement can show.
Running the benchmarks
make check # the fast suite: rungs 1–3, 4a, 5 and the unit tests
make openmp=1 check-slow # rungs 4b and 6, test_rotinv at lmax 128, and three order testsBoth targets run every test, print a summary that names the failing tests, and return a non-zero exit status if any failed. check-slow takes about ten minutes on 32 threads. It raises the stack limit itself, since the 3-D path at \(\ell_\max \ge 128\) needs more than the default 8 MB per thread.
Dumping the comparison data
The benchmark tests can write the numbers behind their comparison. Set the environment variable VILMA_BENCH_DUMP to an existing directory, and each test writes plain-text column tables there (model and reference values on the reference abscissa, with a header line naming the columns):
mkdir -p dump
VILMA_BENCH_DUMP=$PWD/dump bin/test_benchmark_love.x
VILMA_BENCH_DUMP=$PWD/dump bin/test_benchmark_sle.x E2Without the variable the tests write nothing. The figures on the benchmark pages are made from these tables by the scripts in the repository’s validation/ directory.
Per-benchmark pages
- Love numbers: elastic and fluid \(h,\ell,k\) for M3–L70–V01.
- Disc-load response: elastic profiles and the center uplift for the 10° disc.
- Sea-level equation: the five Martinec et al. (2018) cases.
- Rotation: polar motion against Spada test 3/2.
- Low-viscosity zone: lateral viscosity against