Kortwrieg-de Vries Equation FDM Work-Precision Diagrams
Problem Description
The Kortwrieg de Vries (KdV) partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 16,~T = 5$, with the following initial and periodic boundary conditions:
\[\begin{align} \partial_t u(t,x) & = -6u(t,x)\partial_x u(t,x) -\partial_x^3 u(t,x), \\ u(0,x) & = \cos\left(\frac{\pi x}{L}\right), \\ u(t,-L) & = u(t,L) = -1. \end{align}\]
The spatial derivative operators are represented via finite difference approximations on an equispaced grid of points $x_s \in [-L, L]$.
\[\begin{align} \frac{du}{dt} & = -6uD_x u -D_x^3 u, \\ u(0,x_s) & = \cos\left(\frac{\pi x_s}{L}\right), \\ u(t,-L) & = u(t,L) = -1. \end{align}\]
Implementation
using OrdinaryDiffEq
using OrdinaryDiffEqBDF, OrdinaryDiffEqExponentialRK, OrdinaryDiffEqFIRK, OrdinaryDiffEqIMEXMultistep, OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK
using DiffEqDevTools
using SciMLOperators
using LinearSolve
using LinearAlgebra
using SparseArrays
using Sundials
using SummationByPartsOperators
const SBP = SummationByPartsOperators
using Plots
gr();nonlinear_convection!(du, u, p, t) = du .= (-6 * p.alpha / 3) * (u .* (p.D1 * u) + p.D1 * (u .^ 2))
# Construct the problem
function kortwrieg_de_vries(N, L, alpha)
D1 = periodic_derivative_operator(derivative_order = 1, accuracy_order = 4,
xmin = -L, xmax = L, N = N)
D3 = periodic_derivative_operator(derivative_order = 3, accuracy_order = 4,
xmin = -L, xmax = L, N = N)
# u0 = usol.(first(tspan), SBP.grid(D1))
x = SBP.grid(D1)
u0 = @. cos(π * x / L) # Initial condition
p = (; D1, alpha)
tspan = (0.0, 1.0)
prob = SplitODEProblem(MatrixOperator(-p.alpha * sparse(D3)),
nonlinear_convection!,
u0, tspan, p);
return x, prob
end;Reference Solution
Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision.
N = 128 # Number of grid points
L = 16.0 # Domain length
alpha = 5.0 # Time scaling factor
xs, prob = kortwrieg_de_vries(N, L, alpha)
@time sol = solve(prob, AutoVern7(Rodas5P(autodiff=AutoFiniteDiff()));
dt = 1e-4, abstol = 1e-14, reltol = 1e-14);
test_sol = TestSolution(sol) # Reference solution for error estimation
tslices = LinRange(prob.tspan..., 50)
ys = mapreduce(sol, hcat, tslices)
plt = heatmap(xs, tslices, ys', xlabel="x", ylabel="t")9.325016 seconds (10.49 M allocations: 1.096 GiB, 5.39% gc time, 93.77% c
ompilation time)
Work-Precision Diagrams
High Tolerances
Implicit-Explicit Methods
abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [
Dict(:alg => IMEXEuler(), :dts => 1e-4 * multipliers),
Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers),
Dict(:alg => CNLF2(), :dts => 1e-4 * multipliers),
Dict(:alg => SBDF2(), :dts => 1e-4 * multipliers),
]
labels = hcat(
"IMEXEuler",
"CNAB2",
"CNLF2",
"SBDF2",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
print_names=true, names=labels, numruns=5, error_estimate=:l2,
save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
plot(wp, label=labels, markershape=:auto, title="Work-Precision Diagram, High Tolerance")IMEXEuler
CNAB2
CNLF2
SBDF2
106.449272 seconds (63.80 M allocations: 27.695 GiB, 12.99% gc time, 23.51%
compilation time)
Exponential Integrators
abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [
Dict(:alg => NorsettEuler(), :dts => 1e-4 * multipliers),
Dict(:alg => NorsettEuler(krylov=true, m=5), :dts => 1e-4 * multipliers),
Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-4 * multipliers),
Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers),
Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-4 * multipliers),
Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-4 * multipliers)
]
labels = hcat(
"NorsettEuler (caching)",
"NorsettEuler (m=5)",
"NorsettEuler (m=20)",
"ETDRK2 (caching)",
"ETDRK2 (m=5)",
"ETDRK2 (m=20)"
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
print_names=true, names=labels, numruns=5, error_estimate=:l2,
save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, High Tolerance")NorsettEuler (caching)
NorsettEuler (m=5)
NorsettEuler (m=20)
ETDRK2 (caching)
ETDRK2 (m=5)
ETDRK2 (m=20)
250.559218 seconds (128.27 M allocations: 66.499 GiB, 10.45% gc time, 6.64%
compilation time)
Comparisons Between Families
abstols = 0.1 .^ (5:8) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [
Dict(:alg => CNAB2(), :dts => 1e-4 * multipliers),
Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-4 * multipliers),
Dict(:alg => ETDRK2(), :dts => 1e-4 * multipliers),
]
labels = hcat(
"CNAB2 (dense)",
"CNAB2 (Krylov)",
"ETDRK2 (caching)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
print_names=true, names=labels, numruns=5, error_estimate=:l2,
save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances")CNAB2 (dense)
CNAB2 (Krylov)
ETDRK2 (caching)
78.735444 seconds (35.33 M allocations: 32.019 GiB, 18.20% gc time, 4.58%
compilation time)
Low Tolerances
Implicit-Explicit Methods
OrdinaryDiffEq defaults and Sundials Krylov linear solvers.
abstols = 0.1 .^ (8:12)
reltols = 0.1 .^ (5:9)
setups = [
Dict(:alg => KenCarp3()),
Dict(:alg => KenCarp4()),
Dict(:alg => KenCarp5()),
Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:GMRES)),
Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:GMRES)),
Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)),
]
labels = hcat(
"KenCarp3 (default)",
"KenCarp4 (default)",
"KenCarp5 (default)",
"ARKODE3 (Krylov)",
"ARKODE4 (Krylov)",
"ARKODE5 (Krylov)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
print_names=true, names=labels, numruns=5, error_estimate=:l2,
save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Low Tolerances")KenCarp3 (default)
KenCarp4 (default)
KenCarp5 (default)
ARKODE3 (Krylov)
ARKODE4 (Krylov)
ARKODE5 (Krylov)
110.910359 seconds (92.92 M allocations: 14.582 GiB, 5.68% gc time, 36.99%
compilation time)
Exponential Integrators
abstols = 0.1 .^ (7:11) # all fixed dt methods so these don't matter much
reltols = 0.1 .^ (4:8)
multipliers = 0.5 .^ (0:4)
setups = [Dict(:alg => ETDRK3(), :dts => 1e-3 * multipliers),
Dict(:alg => ETDRK4(), :dts => 1e-3 * multipliers),
Dict(:alg => HochOst4(), :dts => 1e-3 * multipliers)]
labels = hcat(
"ETDRK3 (caching)",
"ETDRK4 (caching)",
"HochOst4 (caching)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
print_names=true, names=labels, numruns=5, error_estimate=:l2,
save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
plot(wp, label=labels, markershape=:auto, title="ExpRK Methods, Low Tolerances")ETDRK3 (caching)
ETDRK4 (caching)
HochOst4 (caching)
146.578487 seconds (24.04 M allocations: 45.649 GiB, 4.86% gc time, 6.78% c
ompilation time)
Comparisons Between Families
abstols = 0.1 .^ (7:11)
reltols = 0.1 .^ (4:8)
multipliers = 0.5 .^ (0:4)
setups = [
Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:GMRES)),
Dict(:alg => ETDRK3(), :dts => 1e-2 * multipliers),
Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers),
]
labels = hcat(
"ARKODE5 (Krylov)",
"ETDRK3 (caching)",
"ETDRK4 (caching)",
)
@time wp = WorkPrecisionSet(prob, abstols, reltols, setups;
print_names=true, names=labels, numruns=5, error_estimate=:l2,
save_everystep=false, appxsol=test_sol, maxiters=Int(1e5));
plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances")ARKODE5 (Krylov)
ETDRK3 (caching)
ETDRK4 (caching)
105.731634 seconds (17.07 M allocations: 24.312 GiB, 1.62% gc time, 0.02% c
ompilation time)
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/SimpleHandwrittenPDE","kdv_fdm_wpd.jmd")Computer Information:
Julia Version 1.10.12
Commit d93beab124c (2026-08-15 10:29 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
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (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/SimpleHandwrittenPDE/Project.toml`
[47edcb42] ADTypes v1.24.0
⌃ [2169fc97] AlgebraicMultigrid v2.0.1
[b30e2e7b] ClassicalOrthogonalPolynomials v0.15.20
⌃ [f3b72e0c] DiffEqDevTools v3.4.0
[40713840] IncompleteLU v0.2.1
⌃ [7f56f5a3] LSODA v1.1.0
⌃ [7ed4a6bd] LinearSolve v5.15.0
⌃ [1dea7af3] OrdinaryDiffEq v7.7.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.4
⌃ [bbf590c4] OrdinaryDiffEqCore v4.15.0
⌃ [e0540318] OrdinaryDiffEqExponentialRK v2.3.0
⌃ [5960d6e9] OrdinaryDiffEqFIRK v2.8.0
⌃ [d28bc4f8] OrdinaryDiffEqHighOrderRK v2.2.0
⌃ [9f002381] OrdinaryDiffEqIMEXMultistep v2.2.0
⌅ [d4b830b4] OrdinaryDiffEqMultirate v2.7.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.0
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.0
[358294b1] OrdinaryDiffEqStabilizedRK v2.6.0
[91a5bcdd] Plots v1.41.7
⌃ [31c91b34] SciMLBenchmarks v0.1.3
[c0aeaf25] SciMLOperators v1.30.0
⌃ [9f842d2f] SparseConnectivityTracer v1.2.2
[0a514795] SparseMatrixColorings v0.4.27
⌃ [9f78cca6] SummationByPartsOperators v0.5.96
[c3572dad] Sundials v6.6.0
[37e2e46d] LinearAlgebra
[2f01184e] SparseArrays v1.10.0
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`And the full manifest:
Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/SimpleHandwrittenPDE/Manifest.toml`
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[d7ba0133] Git v1.5.0
[42e2da0e] Grisu v1.0.2
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⌃ [5960d6e9] OrdinaryDiffEqFIRK v2.8.0
⌃ [d28bc4f8] OrdinaryDiffEqHighOrderRK v2.2.0
⌃ [9f002381] OrdinaryDiffEqIMEXMultistep v2.2.0
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⌅ [d4b830b4] OrdinaryDiffEqMultirate v2.7.0
[127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.4
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.0
⌃ [b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.1
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.0
[358294b1] OrdinaryDiffEqStabilizedRK v2.6.0
⌃ [b1df2697] OrdinaryDiffEqTsit5 v2.1.3
⌃ [79d7bb75] OrdinaryDiffEqVerner v2.4.0
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⌅ [aea7be01] PrecompileTools v1.2.1
[21216c6a] Preferences v1.5.2
[43287f4e] PtrArrays v1.4.0
[78ab2635] PureGebal v1.1.0
[0c0d3e7f] PureKLU v1.4.1
[1fd47b50] QuadGK v2.11.3
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[01d81517] RecipesPipeline v0.6.12
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[807425ed] RecurrenceRelationships v0.2.0
⌃ [731186ca] RecursiveArrayTools v4.5.0
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
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⌃ [19f34311] SciMLJacobianOperators v0.1.17
[a6db7da4] SciMLLogging v2.1.0
[c0aeaf25] SciMLOperators v1.30.0
[431bcebd] SciMLPublic v1.3.0
⌃ [53ae85a6] SciMLStructures v1.10.4
[6c6a2e73] Scratch v1.3.0
[f8ebbe35] SemiseparableMatrices v0.4.1
[efcf1570] Setfield v1.1.2
⌃ [992d4aef] Showoff v1.0.3
[777ac1f9] SimpleBufferStream v1.2.0
⌃ [727e6d20] SimpleNonlinearSolve v2.14.0
[ce78b400] SimpleUnPack v1.1.0
[a2af1166] SortingAlgorithms v1.2.3
[a57abbd0] SparseColumnPivotedQR v2.1.7
⌃ [9f842d2f] SparseConnectivityTracer v1.2.2
[0a514795] SparseMatrixColorings v0.4.27
[276daf66] SpecialFunctions v2.9.0
[860ef19b] StableRNGs v1.0.4
[aedffcd0] Static v1.4.6
[0d7ed370] StaticArrayInterface v1.10.0
⌃ [90137ffa] StaticArrays v1.9.19
[1e83bf80] StaticArraysCore v1.4.4
[82ae8749] StatsAPI v1.8.0
[2913bbd2] StatsBase v0.34.13
[4c63d2b9] StatsFuns v2.2.1
[7792a7ef] StrideArraysCore v0.5.9
[69024149] StringEncodings v0.3.7
[09ab397b] StructArrays v0.7.3
⌃ [9f78cca6] SummationByPartsOperators v0.5.96
[c3572dad] Sundials v6.6.0
[2efcf032] SymbolicIndexingInterface v0.3.55
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[62fd8b95] TensorCore v0.1.1
[8290d209] ThreadingUtilities v0.5.6
⌅ [a759f4b9] TimerOutputs v0.5.29
[c751599d] ToeplitzMatrices v0.8.5
[3bb67fe8] TranscodingStreams v0.11.3
[781d530d] TruncatedStacktraces v1.4.0
[5c2747f8] URIs v1.7.0
[3a884ed6] UnPack v1.0.2
[1cfade01] UnicodeFun v0.4.1
[9602ed7d] Unrolled v0.1.5
[41fe7b60] Unzip v0.2.0
[3d5dd08c] VectorizationBase v0.21.74
[81def892] VersionParsing v1.3.0
[44d3d7a6] Weave v0.10.12
[ddb6d928] YAML v0.4.16
[c2297ded] ZMQ v1.5.1
[6e34b625] Bzip2_jll v1.0.9+0
[83423d85] Cairo_jll v1.18.7+0
[ee1fde0b] Dbus_jll v1.16.2+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.8.3+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
[f5851436] FFTW_jll v3.3.12+0
[34b6f7d7] FastTransforms_jll v0.6.4+0
[a3f928ae] Fontconfig_jll v2.17.1+0
[d7e528f0] FreeType2_jll v2.14.3+1
[559328eb] FriBidi_jll v1.0.17+0
⌃ [0656b61e] GLFW_jll v3.4.1+1
⌅ [d2c73de3] GR_jll v0.73.26+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
[020c3dae] Git_LFS_jll v3.7.1+0
[f8c6e375] Git_jll v2.55.0+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
⌅ [2e76f6c2] HarfBuzz_jll v8.5.1+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.1.0+0
[1d63c593] LLVMOpenMP_jll v22.1.7+0
[aae0fff6] LSODA_jll v0.1.2+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
[e7412a2a] Ogg_jll v1.3.6+0
⌅ [656ef2d0] OpenBLAS32_jll v0.3.24+0
[9bd350c2] OpenSSH_jll v10.5.1+0
⌃ [458c3c95] OpenSSL_jll v3.5.7+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
⌃ [36c8627f] Pango_jll v1.58.0+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
⌅ [ca45d3f4] SuiteSparse32_jll v5.10.1+0
[fb77eaff] Sundials_jll v7.5.0+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.3+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
[8f1865be] ZeroMQ_jll v4.3.6+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 v0.17.4+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
[a9144af2] libsodium_jll v1.0.21+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 v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.10.0
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[6462fe0b] Sockets
[2f01184e] SparseArrays v1.10.0
[10745b16] Statistics v1.10.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.1.1+0
[781609d7] GMP_jll v6.2.1+6
[deac9b47] LibCURL_jll v8.4.0+0
[e37daf67] LibGit2_jll v1.6.4+0
[29816b5a] LibSSH2_jll v1.11.0+1
[3a97d323] MPFR_jll v4.2.0+1
[c8ffd9c3] MbedTLS_jll v2.28.1010+0
[14a3606d] MozillaCACerts_jll v2025.12.2
[4536629a] OpenBLAS_jll v0.3.23+5
[05823500] OpenLibm_jll v0.8.5+0
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.2.1+1
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.52.0+1
[3f19e933] p7zip_jll v17.6.1+0
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`