Oceananigans.jl: Surface artifacts in GPU simulations
Running a "minimal icy moon" simulation on the CPU seems fine, but on the GPU it exhibits weird surface artifacts that propagate at the top most vertical level!
CPU:
GPU:
Simulation script: https://github.com/ali-ramadhan/JuicyMoons.jl/blob/ali/minimal-icy-moon/simulation/minimal_icy_moon.jl
Setup description
- Topology: (Periodic, Periodic, Bounded)
- Resolution: 50 x 250 x 25 grid points
- Domain: 50 x 250 x 25 km (isotropic grid spacing of 200 m)
- Boundary conditions: default no normal flow + free slip + no flux (for tracers)
- Gravity: g = 1.3 m/s²
- Closure: inviscid, i.e. isotropic diffusivity with ν = κ = 0
- Coriolis: non-traditional f-plane with fz = fy = 2.97e-05 (45°N)
- Initial conditions: u=v=w=0 and T = exp(-0.2(z+Lz/2)^2) * ε(1e-10) where ε is Gaussian noise with zero mean and variance 1e-10 (so noise is concentrated at mid depth)
ali-ramadhan
All 26 comments
Out of curiosity, what if you use non-traditional f-plane with fz = fy = 0.0?
navidcy
on 12 Nov 2020
After one iteration, the CPU and GPU u and v fields match everywhere except at
- north and south halos for
k=0(bottom halo region) k = Nz(entire level including east, west, north, south halos)k = Nz+1(entire level excluding east, west, north, south halos)
As for the w fields, they match everywhere at k = Nz (entire level including east, west, north, south halos).
It seems like there's a difference between the CPU and GPU in filling north and south halos for u and v at k = 0 (they're corner halos) so this could be a scheduling issue (halos are being filled in different order). But with our current numerical algorithm I don't see how a difference in the bottom corners would lead to different velocities in the topmost vertical level (leaving the interior completely untouched).
The bigger issue seems to be that u, v, w all don't match at k = Nz (entire level including east, west, north, south halos). For this I don't really have any ideas why...
Comparing u
```
julia> u₁_gpu .- u₁_cpu
52×252×27 Array{Float32,3}:
[:, :, 1] =
0.0 -2.1735f-12 1.55519f-11 -1.01398f-11 4.51001f-12 … 7.79226f-12 -5.71063f-12 -1.28999f-11 9.6465f-12 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋱ ⋮
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 1.10783f-11 -1.35082f-11 -1.35387f-11 -8.96711f-12 -7.30399f-12 -6.13967f-12 9.11722f-12 3.24933f-12 0.0
[:, :, 2] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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[:, :, 3] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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...
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 26] =
-5.38578f-15 -5.98418f-15 -6.46774f-15 -6.81876f-15 -7.0128f-15 … -3.88357f-15 -4.6831f-15 -5.38578f-15 -5.98418f-15
-4.2658f-15 -4.69527f-15 -5.00186f-15 -5.1802f-15 -5.21871f-15 -3.04912f-15 -3.71556f-15 -4.2658f-15 -4.69527f-15
-2.88249f-15 -3.12326f-15 -3.23831f-15 -3.23601f-15 -3.1199f-15 -2.00341f-15 -2.50886f-15 -2.88249f-15 -3.12326f-15
-1.2746f-15 -1.31531f-15 -1.23293f-15 -1.04855f-15 -7.81116f-16 -7.76992f-16 -1.09572f-15 -1.2746f-15 -1.31531f-15
5.05878f-16 6.70561f-16 9.5027f-16 1.31363f-15 1.72763f-15 5.85827f-16 4.76639f-16 5.05878f-16 6.70561f-16
2.39589f-15 2.76711f-15 3.2396f-15 3.77484f-15 4.32844f-15 … 2.0302f-15 2.1492f-15 2.39589f-15 2.76711f-15
4.32494f-15 4.8978f-15 5.55234f-15 6.24624f-15 6.92746f-15 3.49978f-15 3.85788f-15 4.32494f-15 4.8978f-15
6.22068f-15 6.97901f-15 7.79412f-15 8.62334f-15 9.41146f-15 4.94801f-15 5.54256f-15 6.22068f-15 6.97901f-15
8.01413f-15 8.92551f-15 9.86513f-15 1.07934f-14 1.16565f-14 6.34235f-15 7.15191f-15 8.01413f-15 8.92551f-15
9.64069f-15 1.06569f-14 1.16715f-14 1.26509f-14 1.35469f-14 7.6551f-15 8.63917f-15 9.64069f-15 1.06569f-14
1.10357f-14 1.21f-14 1.31333f-14 1.41106f-14 1.49922f-14 … 8.84536f-15 9.9506f-15 1.10357f-14 1.21f-14
1.21262f-14 1.31834f-14 1.41822f-14 1.51074f-14 1.59297f-14 9.84349f-15 1.10137f-14 1.21262f-14 1.31834f-14
1.28264f-14 1.3831f-14 1.47528f-14 1.55866f-14 1.63151f-14 1.05502f-14 1.1735f-14 1.28264f-14 1.3831f-14
1.30476f-14 1.39668f-14 1.47831f-14 1.55016f-14 1.61177f-14 1.08554f-14 1.20141f-14 1.30476f-14 1.39668f-14
1.27218f-14 1.35353f-14 1.42319f-14 1.48275f-14 1.53309f-14 1.06699f-14 1.17719f-14 1.27218f-14 1.35353f-14
1.18284f-14 1.25278f-14 1.31041f-14 1.35849f-14 1.39922f-14 … 9.95473f-15 1.09777f-14 1.18284f-14 1.25278f-14
1.04092f-14 1.0997f-14 1.14649f-14 1.1852f-14 1.21925f-14 8.73533f-15 9.6641f-15 1.04092f-14 1.0997f-14
8.56317f-15 9.05042f-15 9.43056f-15 9.75268f-15 1.0061f-14 7.09926f-15 7.9234f-15 8.56317f-15 9.05042f-15
⋮ ⋱ ⋮
-1.32188f-15 -1.29409f-15 -1.32188f-15 -1.45265f-15 -1.72361f-15 … -1.35817f-15 -1.3573f-15 -1.32188f-15 -1.29409f-15
-1.42456f-15 -1.48042f-15 -1.55749f-15 -1.70519f-15 -1.96731f-15 -1.21136f-15 -1.34579f-15 -1.42456f-15 -1.48042f-15
-1.74661f-15 -1.91138f-15 -2.06773f-15 -2.2639f-15 -2.54659f-15 -1.25147f-15 -1.53463f-15 -1.74661f-15 -1.91138f-15
-2.26019f-15 -2.54853f-15 -2.80664f-15 -3.07919f-15 -3.41229f-15 -1.4757f-15 -1.9083f-15 -2.26019f-15 -2.54853f-15
-2.92251f-15 -3.33873f-15 -3.71143f-15 -4.08074f-15 -4.48958f-15 -1.85986f-15 -2.4339f-15 -2.92251f-15 -3.33873f-15
-3.67671f-15 -4.21734f-15 -4.70858f-15 -5.18596f-15 -5.68756f-15 … -2.35834f-15 -3.06083f-15 -3.67671f-15 -4.21734f-15
-4.45944f-15 -5.11623f-15 -5.723f-15 -6.31039f-15 -6.91015f-15 -2.91389f-15 -3.72901f-15 -4.45944f-15 -5.11623f-15
-5.21119f-15 -5.97235f-15 -6.68532f-15 -7.37462f-15 -8.06377f-15 -3.47134f-15 -4.38111f-15 -5.21119f-15 -5.97235f-15
-5.88245f-15 -6.73148f-15 -7.53389f-15 -8.30598f-15 -9.06078f-15 -3.98689f-15 -4.97073f-15 -5.88245f-15 -6.73148f-15
-6.43339f-15 -7.34729f-15 -8.21373f-15 -9.03861f-15 -9.82262f-15 -4.42821f-15 -5.46261f-15 -6.43339f-15 -7.34729f-15
-6.82972f-15 -7.77895f-15 -8.6761f-15 -9.51555f-15 -1.0285f-14 … -4.76772f-15 -5.82699f-15 -6.82972f-15 -7.77895f-15
-7.03857f-15 -7.98919f-15 -8.87894f-15 -9.69122f-15 -1.04019f-14 -4.97568f-15 -6.03373f-15 -7.03857f-15 -7.98919f-15
-7.02675f-15 -7.943f-15 -8.78594f-15 -9.53067f-15 -1.01447f-14 -5.01874f-15 -6.05042f-15 -7.02675f-15 -7.943f-15
-6.7623f-15 -7.60795f-15 -8.36567f-15 -9.00701f-15 -9.49615f-15 -4.86473f-15 -5.84541f-15 -6.7623f-15 -7.60795f-15
-6.21954f-15 -6.95855f-15 -7.595f-15 -8.1029f-15 -8.44924f-15 -4.48989f-15 -5.39382f-15 -6.21954f-15 -6.95855f-15
-5.38578f-15 -5.98418f-15 -6.46774f-15 -6.81876f-15 -7.0128f-15 … -3.88357f-15 -4.6831f-15 -5.38578f-15 -5.98418f-15
-4.2658f-15 -4.69527f-15 -5.00186f-15 -5.1802f-15 -5.21871f-15 -3.04912f-15 -3.71556f-15 -4.2658f-15 -4.69527f-15
[:, :, 27] =
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 -4.69527f-15 -5.00186f-15 -5.1802f-15 -5.21871f-15 -2.27887f-15 -3.04912f-15 -3.71556f-15 -4.2658f-15 0.0
0.0 -3.12326f-15 -3.23831f-15 -3.23601f-15 -3.1199f-15 -1.37991f-15 -2.00341f-15 -2.50886f-15 -2.88249f-15 0.0
0.0 -1.31531f-15 -1.23293f-15 -1.04855f-15 -7.81116f-16 -3.33126f-16 -7.76992f-16 -1.09572f-15 -1.2746f-15 0.0
0.0 6.70561f-16 9.5027f-16 1.31363f-15 1.72763f-15 8.18059f-16 5.85827f-16 4.76639f-16 5.05878f-16 0.0
0.0 2.76711f-15 3.2396f-15 3.77484f-15 4.32844f-15 … 2.02344f-15 2.0302f-15 2.1492f-15 2.39589f-15 0.0
0.0 4.8978f-15 5.55234f-15 6.24624f-15 6.92746f-15 3.23627f-15 3.49978f-15 3.85788f-15 4.32494f-15 0.0
0.0 6.97901f-15 7.79412f-15 8.62334f-15 9.41146f-15 4.42523f-15 4.94801f-15 5.54256f-15 6.22068f-15 0.0
0.0 8.92551f-15 9.86513f-15 1.07934f-14 1.16565f-14 5.57726f-15 6.34235f-15 7.15191f-15 8.01413f-15 0.0
0.0 1.06569f-14 1.16715f-14 1.26509f-14 1.35469f-14 6.684f-15 7.6551f-15 8.63917f-15 9.64069f-15 0.0
0.0 1.21f-14 1.31333f-14 1.41106f-14 1.49922f-14 … 7.71818f-15 8.84536f-15 9.9506f-15 1.10357f-14 0.0
0.0 1.31834f-14 1.41822f-14 1.51074f-14 1.59297f-14 8.61489f-15 9.84349f-15 1.10137f-14 1.21262f-14 0.0
0.0 1.3831f-14 1.47528f-14 1.55866f-14 1.63151f-14 9.27142f-15 1.05502f-14 1.1735f-14 1.28264f-14 0.0
0.0 1.39668f-14 1.47831f-14 1.55016f-14 1.61177f-14 9.56997f-15 1.08554f-14 1.20141f-14 1.30476f-14 0.0
0.0 1.35353f-14 1.42319f-14 1.48275f-14 1.53309f-14 9.41308f-15 1.06699f-14 1.17719f-14 1.27218f-14 0.0
0.0 1.25278f-14 1.31041f-14 1.35849f-14 1.39922f-14 … 8.75476f-15 9.95473f-15 1.09777f-14 1.18284f-14 0.0
0.0 1.0997f-14 1.14649f-14 1.1852f-14 1.21925f-14 7.6156f-15 8.73533f-15 9.6641f-15 1.04092f-14 0.0
0.0 9.05042f-15 9.43056f-15 9.75268f-15 1.0061f-14 6.08029f-15 7.09926f-15 7.9234f-15 8.56317f-15 0.0
⋮ ⋱ ⋮
0.0 -1.29409f-15 -1.32188f-15 -1.45265f-15 -1.72361f-15 … -1.28997f-15 -1.35817f-15 -1.3573f-15 -1.32188f-15 0.0
0.0 -1.48042f-15 -1.55749f-15 -1.70519f-15 -1.96731f-15 -1.0001f-15 -1.21136f-15 -1.34579f-15 -1.42456f-15 0.0
0.0 -1.91138f-15 -2.06773f-15 -2.2639f-15 -2.54659f-15 -8.87338f-16 -1.25147f-15 -1.53463f-15 -1.74661f-15 0.0
0.0 -2.54853f-15 -2.80664f-15 -3.07919f-15 -3.41229f-15 -9.60709f-16 -1.4757f-15 -1.9083f-15 -2.26019f-15 0.0
0.0 -3.33873f-15 -3.71143f-15 -4.08074f-15 -4.48958f-15 -1.20349f-15 -1.85986f-15 -2.4339f-15 -2.92251f-15 0.0
0.0 -4.21734f-15 -4.70858f-15 -5.18596f-15 -5.68756f-15 … -1.57453f-15 -2.35834f-15 -3.06083f-15 -3.67671f-15 0.0
0.0 -5.11623f-15 -5.723f-15 -6.31039f-15 -6.91015f-15 -2.01985f-15 -2.91389f-15 -3.72901f-15 -4.45944f-15 0.0
0.0 -5.97235f-15 -6.68532f-15 -7.37462f-15 -8.06377f-15 -2.48737f-15 -3.47134f-15 -4.38111f-15 -5.21119f-15 0.0
0.0 -6.73148f-15 -7.53389f-15 -8.30598f-15 -9.06078f-15 -2.93596f-15 -3.98689f-15 -4.97073f-15 -5.88245f-15 0.0
0.0 -7.34729f-15 -8.21373f-15 -9.03861f-15 -9.82262f-15 -3.33486f-15 -4.42821f-15 -5.46261f-15 -6.43339f-15 0.0
0.0 -7.77895f-15 -8.6761f-15 -9.51555f-15 -1.0285f-14 … -3.65636f-15 -4.76772f-15 -5.82699f-15 -6.82972f-15 0.0
0.0 -7.98919f-15 -8.87894f-15 -9.69122f-15 -1.04019f-14 -3.86883f-15 -4.97568f-15 -6.03373f-15 -7.03857f-15 0.0
0.0 -7.943f-15 -8.78594f-15 -9.53067f-15 -1.01447f-14 -3.9366f-15 -5.01874f-15 -6.05042f-15 -7.02675f-15 0.0
0.0 -7.60795f-15 -8.36567f-15 -9.00701f-15 -9.49615f-15 -3.8265f-15 -4.86473f-15 -5.84541f-15 -6.7623f-15 0.0
0.0 -6.95855f-15 -7.595f-15 -8.1029f-15 -8.44924f-15 -3.51603f-15 -4.48989f-15 -5.39382f-15 -6.21954f-15 0.0
0.0 -5.98418f-15 -6.46774f-15 -6.81876f-15 -7.0128f-15 … -2.9977f-15 -3.88357f-15 -4.6831f-15 -5.38578f-15 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
```
Comparing v
```
julia> v₁_gpu .- v₁_cpu
52×252×27 Array{Float32,3}:
[:, :, 1] =
0.0 -2.41934f-12 1.93449f-11 -6.02374f-12 2.20067f-12 … 2.8827f-12 -2.44937f-12 -1.01371f-11 3.26672f-12 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋱ ⋮
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 5.40959f-12 -5.24162f-12 -6.05426f-12 6.7723f-12 -9.78057f-13 -1.28505f-12 5.11975f-12 -2.60117f-12 0.0
[:, :, 2] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 3] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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...
[:, :, 25] =
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 26] =
-7.87254f-15 -7.37727f-15 -6.94443f-15 -6.52495f-15 -6.0547f-15 … -8.98131f-15 -8.42959f-15 -7.87254f-15 -7.37727f-15
-8.42278f-15 -7.80674f-15 -7.25102f-15 -6.70329f-15 -6.09321f-15 -9.75155f-15 -9.09603f-15 -8.42278f-15 -7.80674f-15
-8.79641f-15 -8.04751f-15 -7.36607f-15 -6.701f-15 -5.97709f-15 -1.03751f-14 -9.60148f-15 -8.79641f-15 -8.04751f-15
-8.97529f-15 -8.08823f-15 -7.28368f-15 -6.51662f-15 -5.70966f-15 -1.08189f-14 -9.92021f-15 -8.97529f-15 -8.08823f-15
-8.94605f-15 -7.92355f-15 -7.00397f-15 -6.15326f-15 -5.29566f-15 -1.10512f-14 -1.00294f-14 -8.94605f-15 -7.92355f-15
-8.69935f-15 -7.55233f-15 -6.53149f-15 -5.61802f-15 -4.74206f-15 … -1.10444f-14 -9.9104f-15 -8.69935f-15 -7.55233f-15
-8.2323f-15 -6.97946f-15 -5.87695f-15 -4.92412f-15 -4.06084f-15 -1.07809f-14 -9.55231f-15 -8.2323f-15 -6.97946f-15
-7.55418f-15 -6.22113f-15 -5.06183f-15 -4.0949f-15 -3.27273f-15 -1.02581f-14 -8.95776f-15 -7.55418f-15 -6.22113f-15
-6.69196f-15 -5.30975f-15 -4.12222f-15 -3.16664f-15 -2.40962f-15 -9.493f-15 -8.14819f-15 -6.69196f-15 -5.30975f-15
-5.69044f-15 -4.2935f-15 -3.10769f-15 -2.1872f-15 -1.5136f-15 -8.5219f-15 -7.16413f-15 -5.69044f-15 -4.2935f-15
-4.60535f-15 -3.22919f-15 -2.07443f-15 -1.20984f-15 -6.31987f-16 … -7.39472f-15 -6.05889f-15 -4.60535f-15 -3.22919f-15
-3.49293f-15 -2.17197f-15 -1.0756f-15 -2.84639f-16 1.90321f-16 -6.16612f-15 -4.88864f-15 -3.49293f-15 -2.17197f-15
-2.40151f-15 -1.1674f-15 -1.53771f-16 5.49195f-16 9.18759f-16 -4.88734f-15 -3.70384f-15 -2.40151f-15 -1.1674f-15
-1.36799f-15 -2.48253f-16 6.62527f-16 1.2677f-15 1.53482f-15 -3.60191f-15 -2.54512f-15 -1.36799f-15 -2.48253f-16
-4.18068f-16 5.65254f-16 1.3591f-15 1.86334f-15 2.03822f-15 -2.34504f-15 -1.4432f-15 -4.18068f-16 5.65254f-16
4.3266f-16 1.26464f-15 1.93539f-15 2.34417f-15 2.44556f-15 … -1.14508f-15 -4.20275f-16 4.3266f-16 1.26464f-15
1.17774f-15 1.85248f-15 2.40328f-15 2.73124f-15 2.78605f-15 -2.53425f-17 5.08492f-16 1.17774f-15 1.85248f-15
1.81751f-15 2.33974f-15 2.78341f-15 3.05336f-15 3.09437f-15 9.93624f-16 1.33263f-15 1.81751f-15 2.33974f-15
⋮ ⋱ ⋮
2.74677f-15 2.26857f-15 1.9072f-15 1.70854f-15 1.68497f-15 … 3.76985f-15 3.27342f-15 2.74677f-15 2.26857f-15
2.66801f-15 2.21271f-15 1.83013f-15 1.56084f-15 1.42285f-15 3.5586f-15 3.13898f-15 2.66801f-15 2.21271f-15
2.45603f-15 2.04794f-15 1.67378f-15 1.36467f-15 1.14016f-15 3.19446f-15 2.85582f-15 2.45603f-15 2.04794f-15
2.10415f-15 1.75959f-15 1.41568f-15 1.09212f-15 8.07055f-16 2.67947f-15 2.42322f-15 2.10415f-15 1.75959f-15
1.61554f-15 1.34337f-15 1.04298f-15 7.2282f-16 3.98209f-16 2.0231f-15 1.84918f-15 1.61554f-15 1.34337f-15
9.99668f-16 8.02736f-16 5.51733f-16 2.4544f-16 -1.03383f-16 … 1.23929f-15 1.14669f-15 9.99668f-16 8.02736f-16
2.69239f-16 1.45948f-16 -5.50319f-17 -3.41954f-16 -7.03139f-16 3.45254f-16 3.31564f-16 2.69239f-16 1.45948f-16
-5.60843f-16 -6.1521f-16 -7.68002f-16 -1.03126f-15 -1.39229f-15 -6.38716f-16 -5.78203f-16 -5.60843f-16 -6.1521f-16
-1.47256f-15 -1.46423f-15 -1.57042f-15 -1.80334f-15 -2.14709f-15 -1.68965f-15 -1.56204f-15 -1.47256f-15 -1.46423f-15
-2.44335f-15 -2.37813f-15 -2.43686f-15 -2.62822f-15 -2.9311f-15 -2.78299f-15 -2.59645f-15 -2.44335f-15 -2.37813f-15
-3.44608f-15 -3.32735f-15 -3.33401f-15 -3.46768f-15 -3.70051f-15 … -3.89436f-15 -3.65572f-15 -3.44608f-15 -3.32735f-15
-4.45092f-15 -4.27797f-15 -4.22376f-15 -4.27996f-15 -4.41119f-15 -5.00121f-15 -4.71377f-15 -4.45092f-15 -4.27797f-15
-5.42725f-15 -5.19421f-15 -5.0667f-15 -5.0247f-15 -5.02517f-15 -6.08335f-15 -5.74545f-15 -5.42725f-15 -5.19421f-15
-6.34414f-15 -6.03986f-15 -5.82442f-15 -5.66604f-15 -5.51431f-15 -7.12158f-15 -6.72613f-15 -6.34414f-15 -6.03986f-15
-7.16986f-15 -6.77887f-15 -6.46087f-15 -6.17394f-15 -5.86065f-15 -8.09544f-15 -7.63006f-15 -7.16986f-15 -6.77887f-15
-7.87254f-15 -7.37727f-15 -6.94443f-15 -6.52495f-15 -6.0547f-15 … -8.98131f-15 -8.42959f-15 -7.87254f-15 -7.37727f-15
-8.42278f-15 -7.80674f-15 -7.25102f-15 -6.70329f-15 -6.09321f-15 -9.75155f-15 -9.09603f-15 -8.42278f-15 -7.80674f-15
[:, :, 27] =
0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0
0.0 -7.80674f-15 -7.25102f-15 -6.70329f-15 -6.09321f-15 -1.02572f-14 -9.75155f-15 -9.09603f-15 -8.42278f-15 0.0
0.0 -8.04751f-15 -7.36607f-15 -6.701f-15 -5.97709f-15 -1.09717f-14 -1.03751f-14 -9.60148f-15 -8.79641f-15 0.0
0.0 -8.08823f-15 -7.28368f-15 -6.51662f-15 -5.70966f-15 -1.15115f-14 -1.08189f-14 -9.92021f-15 -8.97529f-15 0.0
0.0 -7.92355f-15 -7.00397f-15 -6.15326f-15 -5.29566f-15 -1.18373f-14 -1.10512f-14 -1.00294f-14 -8.94605f-15 0.0
0.0 -7.55233f-15 -6.53149f-15 -5.61802f-15 -4.74206f-15 … -1.19147f-14 -1.10444f-14 -9.9104f-15 -8.69935f-15 0.0
0.0 -6.97946f-15 -5.87695f-15 -4.92412f-15 -4.06084f-15 -1.17213f-14 -1.07809f-14 -9.55231f-15 -8.2323f-15 0.0
0.0 -6.22113f-15 -5.06183f-15 -4.0949f-15 -3.27273f-15 -1.12514f-14 -1.02581f-14 -8.95776f-15 -7.55418f-15 0.0
0.0 -5.30975f-15 -4.12222f-15 -3.16664f-15 -2.40962f-15 -1.05188f-14 -9.493f-15 -8.14819f-15 -6.69196f-15 0.0
0.0 -4.2935f-15 -3.10769f-15 -2.1872f-15 -1.5136f-15 -9.55642f-15 -8.5219f-15 -7.16413f-15 -5.69044f-15 0.0
0.0 -3.22919f-15 -2.07443f-15 -1.20984f-15 -6.31987f-16 … -8.41011f-15 -7.39472f-15 -6.05889f-15 -4.60535f-15 0.0
0.0 -2.17197f-15 -1.0756f-15 -2.84639f-16 1.90321f-16 -7.13163f-15 -6.16612f-15 -4.88864f-15 -3.49293f-15 0.0
0.0 -1.1674f-15 -1.53771f-16 5.49195f-16 9.18759f-16 -5.77131f-15 -4.88734f-15 -3.70384f-15 -2.40151f-15 0.0
0.0 -2.48253f-16 6.62527f-16 1.2677f-15 1.53482f-15 -4.37417f-15 -3.60191f-15 -2.54512f-15 -1.36799f-15 0.0
0.0 5.65254f-16 1.3591f-15 1.86334f-15 2.03822f-15 -2.97899f-15 -2.34504f-15 -1.4432f-15 -4.18068f-16 0.0
0.0 1.26464f-15 1.93539f-15 2.34417f-15 2.44556f-15 … -1.61935f-15 -1.14508f-15 -4.20275f-16 4.3266f-16 0.0
0.0 1.85248f-15 2.40328f-15 2.73124f-15 2.78605f-15 -3.25482f-16 -2.53425f-17 5.08492f-16 1.17774f-15 0.0
0.0 2.33974f-15 2.78341f-15 3.05336f-15 3.09437f-15 8.73927f-16 9.93624f-16 1.33263f-15 1.81751f-15 0.0
⋮ ⋱ ⋮
0.0 2.26857f-15 1.9072f-15 1.70854f-15 1.68497f-15 … 4.15718f-15 3.76985f-15 3.27342f-15 2.74677f-15 0.0
0.0 2.21271f-15 1.83013f-15 1.56084f-15 1.42285f-15 3.85661f-15 3.5586f-15 3.13898f-15 2.66801f-15 0.0
0.0 2.04794f-15 1.67378f-15 1.36467f-15 1.14016f-15 3.4118f-15 3.19446f-15 2.85582f-15 2.45603f-15 0.0
0.0 1.75959f-15 1.41568f-15 1.09212f-15 8.07055f-16 2.82436f-15 2.67947f-15 2.42322f-15 2.10415f-15 0.0
0.0 1.34337f-15 1.04298f-15 7.2282f-16 3.98209f-16 2.10238f-15 2.0231f-15 1.84918f-15 1.61554f-15 0.0
0.0 8.02736f-16 5.51733f-16 2.4544f-16 -1.03383f-16 … 1.25851f-15 1.23929f-15 1.14669f-15 9.99668f-16 0.0
0.0 1.45948f-16 -5.50319f-17 -3.41954f-16 -7.03139f-16 3.09663f-16 3.45254f-16 3.31564f-16 2.69239f-16 0.0
0.0 -6.1521f-16 -7.68002f-16 -1.03126f-15 -1.39229f-15 -7.23149f-16 -6.38716f-16 -5.78203f-16 -5.60843f-16 0.0
0.0 -1.46423f-15 -1.57042f-15 -1.80334f-15 -2.14709f-15 -1.81615f-15 -1.68965f-15 -1.56204f-15 -1.47256f-15 0.0
0.0 -2.37813f-15 -2.43686f-15 -2.62822f-15 -2.9311f-15 -2.94524f-15 -2.78299f-15 -2.59645f-15 -2.44335f-15 0.0
0.0 -3.32735f-15 -3.33401f-15 -3.46768f-15 -3.70051f-15 … -4.08838f-15 -3.89436f-15 -3.65572f-15 -3.44608f-15 0.0
0.0 -4.27797f-15 -4.22376f-15 -4.27996f-15 -4.41119f-15 -5.22683f-15 -5.00121f-15 -4.71377f-15 -4.45092f-15 0.0
0.0 -5.19421f-15 -5.0667f-15 -5.0247f-15 -5.02517f-15 -6.34447f-15 -6.08335f-15 -5.74545f-15 -5.42725f-15 0.0
0.0 -6.03986f-15 -5.82442f-15 -5.66604f-15 -5.51431f-15 -7.42583f-15 -7.12158f-15 -6.72613f-15 -6.34414f-15 0.0
0.0 -6.77887f-15 -6.46087f-15 -6.17394f-15 -5.86065f-15 -8.45364f-15 -8.09544f-15 -7.63006f-15 -7.16986f-15 0.0
0.0 -7.37727f-15 -6.94443f-15 -6.52495f-15 -6.0547f-15 … -9.40652f-15 -8.98131f-15 -8.42959f-15 -7.87254f-15 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
```
Comparing w
```
julia> w₁_gpu .- w₁_cpu
52×252×28 Array{Float32,3}:
[:, :, 1] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
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[:, :, 2] =
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⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
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[:, :, 3] =
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⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮
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...
[:, :, 26] =
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1.50727f-15 2.07253f-15 3.43162f-15 5.29497f-15 7.33318f-15 3.36854f-15 1.92534f-15 1.50727f-15 2.07253f-15
1.33357f-14 1.46003f-14 1.65357f-14 1.88799f-14 2.13254f-14 … 1.33233f-14 1.2903f-14 1.33357f-14 1.46003f-14
2.37448f-14 2.55973f-14 2.80006f-14 3.07318f-14 3.35179f-14 2.20586f-14 2.25671f-14 2.37448f-14 2.55973f-14
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⋮ ⋱ ⋮
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```
ali-ramadhan
on 12 Nov 2020
Good idea @navidcy. Looks like it's a problem with the regular f-plane too:
CPU FPlane
GPU FPlane
ali-ramadhan
on 12 Nov 2020
Actually my idea was targeting to see whether the problem was stemming from the non-traditional terms. That's why I said choose NON-traditional Coriolis but set fz and fy to zero.
But from what you've shown (e.g., with the f-plane comparison) seems like the problem lies deeper... :s
navidcy
on 12 Nov 2020
@ali-ramadhan, perhaps change the issue title as it only mentions non-traditional f-plane now.
navidcy
on 12 Nov 2020
Actually it's a problem without any rotation too! Except now the artifacts don't grow or propagate.
Maybe the CPU and GPU pressure solvers aren't doing exactly the same thing? Effect/error must be very small though as regression tests would catch differences between CPU and GPU simulations.
CPU coriolis = nothing
GPU coriolis = nothing
ali-ramadhan
on 12 Nov 2020
(and none of the tests caught this...?)
!!
navidcy
on 12 Nov 2020
Can you make a MWE? (Lower resolution... simplify initial condition)?
navidcy
on 12 Nov 2020
This should run in ~2.5 minutes on a single CPU core: https://github.com/ali-ramadhan/JuicyMoons.jl/blob/ali/minimal-icy-moon/simulation/minimal_icy_moon.jl
Could be made more minimal now that we know the problem is pretty fundamental but I can look at minimizing it further in the morning.
Plotting takes time so I do it using 24 cores: https://github.com/ali-ramadhan/JuicyMoons.jl/blob/ali/minimal-icy-moon/simulation/plot_minimal_icy_moon.py (inb4 anyone judges me because separate Python script, I'm working on getting GeoData.jl + plotting recipes working!)
ali-ramadhan
on 12 Nov 2020
I don't think we have regression tests for channel geometries.
glwagner
on 12 Nov 2020
It this bug related to the channel geometry?
If you are sure for that then let's modify the issue's title...
navidcy
on 12 Nov 2020
Ah sorry --- I was confused because in an earlier conversation we thought this issue might've had to do with the channel, but now I see that it's not.
glwagner
on 12 Nov 2020
It seems like there's a difference between the CPU and GPU in filling north and south halos for u and v at
k = 0(they're corner halos) so this could be a scheduling issue (halos are being filled in different order).
I think @ali-ramadhan is right that this isn't the issue. The corner u point does not actually come into play with these physics, I don't think. In other words, with non-traditional Coriolis we do a 4-point interpolation to get u into w locations. But u[k=0] would only be used for the tendency for w[k=1]; which is the point _on_ the boundary and thus is not used.
glwagner
on 12 Nov 2020
@ali-ramadhan not sure if it will help but one thing to try is ensuring the mean vertical velocity is zero for the first time-step. The pressure solver _should_ eliminate such a feature.
We also might try removing the initial divergence in the velocity field using the technique discussed in #1027 .
glwagner
on 12 Nov 2020
Why any of these issues will be an issue on GPU and not on CPU though?
(probably I just don't understand the gpu-inner-workings of the code...?)
navidcy
on 12 Nov 2020
The CPU and GPU use different pressure solvers. That's why, for example, we don't have a pressure solver for "box" (fully enclosed) geometries on the GPU.
glwagner
on 12 Nov 2020
At the first time step, the horizontally averaged w on both the CPU and GPU is well below machine epsilon, so the pressure solver seems to be doing it's job.
The big difference seems to be that on the CPU the horizontally averaged w at k=Nz is non-zero, while on the GPU it is exactly zero.
Looking at the difference between the CPU and GPU horizontally averaged w, they only differ at k=Nz.
CPU w
julia> w_interior_cpu = ds_cpu["w"][2:end-1, 2:end-1, 2:end-1, 2];
julia> size(w_interior_cpu)
(50, 250, 26)
julia> mean(abs, w_interior_cpu)
2.08642f-13
julia> eps(maximum(abs, w_interior_cpu))
8.6736174f-19
julia> w_avg_cpu = mean(w_interior_cpu, dims=(1, 2))[:]
26-element Array{Float32,1}:
0.0
3.2526064f-24
1.951564f-23
-6.505213f-24
7.1557346f-23
5.2041703f-23
-5.2041703f-23
-7.806256f-23
-5.2041703f-23
-2.0816681f-22
3.4694469f-22
-2.7755575f-22
-8.3266725f-22
-8.3266725f-22
-2.7755575f-22
3.4694469f-22
-2.0816681f-22
-5.2041703f-23
-7.806256f-23
-5.2041703f-23
5.2041703f-23
7.1557346f-23
-6.505213f-24
1.951564f-23
3.2526064f-24
0.0
GPU w
julia> w_interior_gpu = ds_gpu["w"][2:end-1, 2:end-1, 2:end-1, 2];
julia> size(w_interior_gpu)
(50, 250, 26)
julia> mean(abs, w_interior_gpu)
2.109486f-13
julia> eps(maximum(abs, w_interior_gpu))
8.6736174f-19
julia> w_avg_gpu = mean(w_interior_gpu, dims=(1, 2))[:]
26-element Array{Float32,1}:
0.0
3.2526064f-24
1.951564f-23
-6.505213f-24
7.1557346f-23
5.2041703f-23
-5.2041703f-23
-7.806256f-23
-5.2041703f-23
-2.0816681f-22
3.4694469f-22
-2.7755575f-22
-8.3266725f-22
-8.3266725f-22
-2.7755575f-22
3.4694469f-22
-2.0816681f-22
-5.2041703f-23
-7.806256f-23
-5.2041703f-23
5.2041703f-23
7.1557346f-23
-6.505213f-24
1.951564f-23
0.0
0.0
GPU w - CPU w
julia> w_avg_gpu - w_avg_cpu
26-element Array{Float32,1}:
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-3.2526064f-24
0.0
The big difference seems to be that on the GPU the horizontally averaged w at
k=Nzis non-zero, while on the CPU it is exactly zero.
It looks like it's exactly zero on the GPU, not the CPU, or am I reading your post wrong?
glwagner
on 12 Nov 2020
Sorry yes I corrected it. It now reads
The big difference seems to be that on the CPU the horizontally averaged w at
k=Nzis non-zero, while on the GPU it is exactly zero.
ali-ramadhan
on 12 Nov 2020
@ali-ramadhan what's the status of this? I've forgotten about it but I have to admit. But shouldn't we resolve this asap?
navidcy
on 26 Nov 2020
I agree Navid. There's nice science application that awaits this fix,
looking at convection on a 'deep' beta-plane. John
On Thu, Nov 26, 2020, 3:02 PM Navid C. Constantinou <
[email protected]> wrote:
@ali-ramadhan https://github.com/ali-ramadhan what's the status of
this? I've forgotten about it but I have to admit. But shouldn't we resolve
this asap?—
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johncmarshall54
on 26 Nov 2020
Sorry been a little busy with other issues but agree this is a pretty high priority.
I was going to set up a 1D minimal working example reproducing this bug at https://github.com/ali-ramadhan/TransformPlayground.jl which should help debug it if it can be reproduced in its purest form.
ali-ramadhan
on 26 Nov 2020
I ran the convergence tests on the GPU and some of them did not pass (the ones that involved a pressure solve along a wall-bounded dimension): https://github.com/CliMA/Oceananigans.jl/pull/1223#issuecomment-734821927
So the issue is maybe more serious than I first thought.
Perhaps it's obscured by the fact that most of the simulations and tests we do are heavily dominated by advection? Either way, it probably doesn't matter as the bug should be fixed ASAP.
ali-ramadhan
on 27 Nov 2020
I know nothing about GPU's but agree this is all troubling and would be good to fix asap.
If the problem seems to occur without rotation and in a bounded domain because of a pressure solve, I wonder whether a 1D test with say 100 grid points might be able to reproduce the problem, like @ali-ramadhan suggested?
Using different pressure solvers for CPU's and GPU's is okay, and probably encouraged and some solvers work faster on certain architectures, but can we set the tolerance in each case? I would think that we want that. If the GPU is giving us a very fast answer but not as accurate as we want, then I think we would all agree that this is a problem.
francispoulin
on 27 Nov 2020
Another thought. I was reading how the MITgcm sets up the boundary conditions for the pressure solver here. The docs discuss how the boundary condition in terms of the nonhydrostatic pressure is an inhomogeneous Neumann (derivative) in the vertical at the surface, but can be rewritten as a homogeneous Neumann (derivative) condition using a source-charge, a la Williams 1969.
I looked at where the code solves the pressure problem here and it seems to be solving a homogeneous Neumann (derivative) boundary condition for ressure. It's doing the same thing for GPU's as well as CPU's, so this doesn't explain any there is a difference, but can someone tell me if this source-charge method is used to make the boundary condition homogeneous?
francispoulin
on 28 Nov 2020
@francispoulin you are correct that a source charge is added to the Poisson RHS to facilitate a solve using eigenfunctions for the homogeneous Neumann problem.
The way the source charge is added is very subtle: the source is added by setting the predictor velocity boundary values (typically this means zeroing them out, when the boundary is impermeable):
This algorithm is very subtle because one can implement it by simply _not updating the predictor velocities_ on boundaries. Changing the wall-normal values of the predictor velocity field modifies the predictor velocity divergence that contributes to the RHS of the pressure Poisson equation. The reason this is an effective source term is because the predictor velocities _do not_ satisfy the same boundary conditions as the physical velocity field (in particular, the predictor velocities have non-zero wall normal components when there is a pressure gradient on the boundary). Thus zeroing out the wall-normal predictor velocities changes the Poisson equation RHS. With some head scratching, it turns out that the modification is precisely what is needed to describe pressure gradients on the boundary.
We need to write this up somewhere, not least because there's no clear reference for this algorithm in the literature.
We also would like to generalize the algorithm to work for time-varying wall-normal velocities (a separate issue but worth noting here that the current algorithm does not work for this case).
glwagner
on 29 Nov 2020
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