Acceleration function benchmarks

Solving the equations of notions for an N-body problem implies solving a (large) system of differential equations. In DifferentialEquations.jl these are represented through ODE or SDE problems. To build the problem we need a function that describe the equations. In the case of N-body problems, this function gives the accelerations for the particles in the system.

Here we will test the performance of several acceleration functions used in N-body simulations. The systems that will be used are not necessarily realistic as we are not solving the problem, we just time how fast is an acceleration function call.

using BenchmarkTools, NBodySimulator
using NBodySimulator: gather_bodies_initial_coordinates,
                      gather_accelerations_for_potentials,
                      gather_simultaneous_acceleration, gather_group_accelerations
using StaticArrays

const SUITE = BenchmarkGroup();

function acceleration(simulation)
    (u0, v0, n) = gather_bodies_initial_coordinates(simulation)

    acceleration_functions = gather_accelerations_for_potentials(simulation)
    simultaneous_acceleration = gather_simultaneous_acceleration(simulation)

    function soode_system!(dv, v, u, p, t)
        @inbounds for i in 1:n
            a = MVector(0.0, 0.0, 0.0)
            for acceleration! in acceleration_functions
                acceleration!(a, u, v, t, i);
            end
            dv[:, i] .= a
        end
        for acceleration! in simultaneous_acceleration
            acceleration!(dv, u, v, t);
        end
    end

    return soode_system!
end

acceleration (generic function with 1 method)

Gravitational potential

let SUITE=SUITE
    G = 6.67e-11 # m^3/kg/s^2
    N = 200 # number of bodies/particles
    m = 1.0 # mass of each of them
    v = 10.0 # mean velocity
    L = 20.0 # size of the cell side

    bodies = generate_bodies_in_cell_nodes(N, m, v, L)
    g_parameters = GravitationalParameters(G)
    system = PotentialNBodySystem(bodies, Dict(:gravitational => g_parameters))
    tspan = (0.0, 1.0)
    simulation = NBodySimulation(system, tspan)

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $g_parameters, 0.0) setup=(dv=zero($v0)) evals=1

    SUITE["gravitational"] = b
end

Benchmark(evals=1, seconds=5.0, samples=10000)

Coulomb potential

let SUITE=SUITE
    n = 200
    bodies = ChargedParticle[]
    L = 20.0
    m = 1.0
    q = 1.0
    count = 1
    dL = L / (ceil(n^(1 / 3)) + 1)
    for x in (dL / 2):dL:L, y in (dL / 2):dL:L, z in (dL / 2):dL:L
        if count > n
            break
        end
        r = SVector(x, y, z)
        v = SVector(0.0, 0.0, 0.0)
        body = ChargedParticle(r, v, m, q)
        push!(bodies, body)
        count += 1
    end

    k = 9e9
    τ = 0.01 * dL / sqrt(2 * k * q * q / (dL * m))
    t1 = 0.0
    t2 = 1000 * τ

    potential = ElectrostaticParameters(k, 0.45 * L)
    system = PotentialNBodySystem(bodies, Dict(:electrostatic => potential))
    pbc = CubicPeriodicBoundaryConditions(L)
    simulation = NBodySimulation(system, (t1, t2), pbc)

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $potential, 0.0) setup=(dv=zero($v0)) evals=1

    SUITE["coulomb"] = b
end

Benchmark(evals=1, seconds=5.0, samples=10000)

Magnetic dipole potential

let SUITE=SUITE
    n = 200
    bodies = MagneticParticle[]
    L = 20.0
    m = 1.0
    count = 1
    dL = L / (ceil(n^(1 / 3)) + 1)
    for x in (dL / 2):dL:L, y in (dL / 2):dL:L, z in (dL / 2):dL:L
        if count > n
            break
        end
        r = SVector(x, y, z)
        v = SVector(0.0, 0.0, 0.0)
        mm = rand(SVector{3})
        body = MagneticParticle(r, v, m, mm)
        push!(bodies, body)
        count += 1
    end

    μ_4π = 1e-7
    t1 = 0.0  # s
    t2 = 1.0 # s
    τ = (t2 - t1) / 100

    parameters = MagnetostaticParameters(μ_4π)
    system = PotentialNBodySystem(bodies, Dict(:magnetic => parameters))
    simulation = NBodySimulation(system, (t1, t2))

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $parameters, 0.0) setup=(dv=zero($v0)) evals=1

    SUITE["magnetic_dipole"] = b
end

Benchmark(evals=1, seconds=5.0, samples=10000)

Lennard Jones potential

let SUITE=SUITE
    T = 120.0 # K
    T0 = 90.0 # K
    kb = 8.3144598e-3 # kJ/(K*mol)
    ϵ = T * kb
    σ = 0.34 # nm
    ρ = 1374/1.6747# Da/nm^3
    N = 200
    m = 39.95# Da = 216 # number of bodies/particles
    L = (m*N/ρ)^(1/3)#10.229σ
    R = 0.5*L
    v_dev = sqrt(kb * T / m)
    bodies = generate_bodies_in_cell_nodes(N, m, v_dev, L)

    τ = 0.5e-3 # ps or 1e-12 s
    t1 = 0.0
    t2 = 2000τ

    lj_parameters = LennardJonesParameters(ϵ, σ, R)
    lj_system = PotentialNBodySystem(bodies, Dict(:lennard_jones => lj_parameters));

    pbc = CubicPeriodicBoundaryConditions(L)
    simulation = NBodySimulation(lj_system, (t1, t2), pbc, kb)

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $lj_parameters, 0.0) setup=(dv=zero($v0)) evals=1

    SUITE["lennard_jones"] = b
end

Benchmark(evals=1, seconds=5.0, samples=10000)

WaterSPCFw model

function acceleration(simulation::NBodySimulation{<:WaterSPCFw})
    (u0, v0, n) = gather_bodies_initial_coordinates(simulation)

    (o_accelerations, h_accelerations) = gather_accelerations_for_potentials(simulation)
    group_accelerations = gather_group_accelerations(simulation)
    simultaneous_acceleration = gather_simultaneous_acceleration(simulation)

    function soode_system!(dv, v, u, p, t)
        @inbounds for i in 1:n
            a = MVector(0.0, 0.0, 0.0)
            for acceleration! in o_accelerations
                acceleration!(a, u, v, t, 3 * (i - 1) + 1);
            end
            dv[:, 3 * (i - 1) + 1] .= a
        end
        @inbounds for i in 1:n, j in (2, 3)

            a = MVector(0.0, 0.0, 0.0)
            for acceleration! in h_accelerations
                acceleration!(a, u, v, t, 3 * (i - 1) + j);
            end
            dv[:, 3 * (i - 1) + j] .= a
        end
        @inbounds for i in 1:n
            for acceleration! in group_accelerations
                acceleration!(dv, u, v, t, i);
            end
        end
        for acceleration! in simultaneous_acceleration
            acceleration!(dv, u, v, t);
        end
    end

    return soode_system!
end

let SUITE=SUITE
    T = 370 # K
    T0 = 275 # K
    kb = 8.3144598e-3 # kJ/(K*mol)
    ϵOO = 0.1554253*4.184 # kJ
    σOO = 0.3165492 # nm
    ρ = 997/1.6747# Da/nm^3
    mO = 15.999 # Da
    mH = 1.00794 # Da
    mH2O = mO+2*mH
    N = 200
    L = (mH2O*N/ρ)^(1/3)
    R = 0.9 # ~3*σOO
    Rel = 0.49*L
    v_dev = sqrt(kb * T / mH2O)
    τ = 0.5e-3 # ps
    t1 = 0τ
    t2 = 5τ # ps
    k_bond = 1059.162*4.184*1e2 # kJ/(mol*nm^2)
    k_angle = 75.90*4.184 # kJ/(mol*rad^2)
    rOH = 0.1012 # nm
    ∠HOH = 113.24*pi/180 # rad
    qH = 0.41
    qO = -0.82
    k = 138.935458 #
    bodies = generate_bodies_in_cell_nodes(N, mH2O, v_dev, L)
    jl_parameters = LennardJonesParameters(ϵOO, σOO, R)
    e_parameters = ElectrostaticParameters(k, Rel)
    spc_parameters = SPCFwParameters(rOH, ∠HOH, k_bond, k_angle)
    pbc = CubicPeriodicBoundaryConditions(L)
    water = WaterSPCFw(bodies, mH, mO, qH, qO, jl_parameters, e_parameters, spc_parameters);
    simulation = NBodySimulation(water, (t1, t2), pbc, kb);

    f = acceleration(simulation)
    u0, v0, n = gather_bodies_initial_coordinates(simulation)
    dv = zero(v0)

    b = @benchmarkable $f(dv, $v0, $u0, $spc_parameters, 0.0) setup=(dv=zero($v0)) evals=1

    SUITE["water_spcfw"] = b
end

Benchmark(evals=1, seconds=5.0, samples=10000)

Here are the results of the benchmarks

r = run(SUITE)

minimum(r)

5-element BenchmarkTools.BenchmarkGroup:
  tags: []
  "gravitational" => TrialEstimate(14.544 ms)
  "coulomb" => TrialEstimate(908.996 μs)
  "lennard_jones" => TrialEstimate(643.426 μs)
  "water_spcfw" => TrialEstimate(9.927 ms)
  "magnetic_dipole" => TrialEstimate(40.696 ms)

and

memory(r)

5-element BenchmarkTools.BenchmarkGroup:
  tags: []
  "gravitational" => 8944000
  "coulomb" => 12800
  "lennard_jones" => 12800
  "water_spcfw" => 108912
  "magnetic_dipole" => 26796800

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/NBodySimulator","acceleration_functions.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/NBodySimulator/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
  [a93c6f00] DataFrames v1.8.2
  [0e6f8da7] NBodySimulator v1.16.0
  [1dea7af3] OrdinaryDiffEq v7.8.1
  [af6ede74] OrdinaryDiffEqRKN v2.2.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v2.2.2
  [91a5bcdd] Plots v1.41.7
  [33c8b6b6] ProgressLogging v0.1.6
  [31c91b34] SciMLBenchmarks v0.2.1
  [90137ffa] StaticArrays v1.9.20
  [f3b207a7] StatsPlots v0.15.8

And the full manifest:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/NBodySimulator/Manifest.toml`
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  [38a345b3] Libuuid_jll v2.42.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.58.2+0
  [30392449] Pixman_jll v0.46.4+0
  [c0090381] Qt6Base_jll v6.10.2+2
  [629bc702] Qt6Declarative_jll v6.10.2+2
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
  [f50d1b31] Rmath_jll v0.5.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.4+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.8+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [1a1011a3] SharedArrays v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`