Nonlinear Schrödinger Equation Pseudospectral Methods Work-Precision Diagrams

using OrdinaryDiffEq
using OrdinaryDiffEqBDF, OrdinaryDiffEqExponentialRK, OrdinaryDiffEqFIRK, OrdinaryDiffEqHighOrderRK, OrdinaryDiffEqIMEXMultistep, OrdinaryDiffEqRosenbrock
using DiffEqDevTools
using SciMLOperators
using LinearSolve
using LinearAlgebra
using SparseArrays
using Sundials
using SummationByPartsOperators
const SBP = SummationByPartsOperators
using Plots
gr()

Plots.GRBackend()

Problem Description

The focusing cubic nonlinear Schrödinger (NLS) equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times \mathbb R,~L = 16,~T = 1$, with periodic boundary conditions:

\[\begin{align} i\partial_t \psi(t,x) + \alpha\partial_x^2 \psi(t,x) + 2|\psi(t,x)|^2 \psi(t,x) &= 0, \\ \psi(0,x) &= \cos\left(\frac{\pi x}{L}\right), \\ \psi(t,-L) &= \psi(t,L). \end{align}\]

Splitting into real and imaginary parts $\psi = u + iv$:

\[\begin{align} \partial_t u &= -\alpha\partial_x^2 v - 2(u^2 + v^2)v, \\ \partial_t v &= \alpha\partial_x^2 u + 2(u^2 + v^2)u. \end{align}\]

The spatial derivative operators are represented via Fourier pseudospectral approximations. The linear operator $\alpha\begin{bmatrix} 0 & -D_x^2 \\ D_x^2 & 0 \end{bmatrix}$ has purely imaginary eigenvalues, reflecting the dispersive (not dissipative) nature of the NLS equation.

Note: Unlike the FDM version, the spectral differentiation matrix $D_x^2$ is dense, so the Jacobian is also dense. Sparse Jacobian techniques do not apply here. For sparse Jacobian stiff method comparisons, see the FDM benchmark.

Implementation

function nls_nonlinear!(du, w, p, t)
    N = p.N
    u = @view w[1:N]
    v = @view w[N+1:2*N]
    du_u = @view du[1:N]
    du_v = @view du[N+1:2*N]
    @. du_u = -2 * (u^2 + v^2) * v
    @. du_v =  2 * (u^2 + v^2) * u
end

function nonlinear_schrodinger(N, L, alpha)
    D1 = fourier_derivative_operator(xmin = -L, xmax = L, N = N)
    D2 = D1^2  # Second derivative via squaring first derivative
    x = SBP.grid(D1)

    D2_mat = Matrix(D2)
    Z = zeros(N, N)
    A = alpha * [Z  -D2_mat;
                 D2_mat  Z]

    u0 = @. cos(π * x / L)
    v0 = zeros(N)
    w0 = [u0; v0]

    p = (; N)
    tspan = (0.0, 1.0)
    prob = SplitODEProblem(MatrixOperator(A), nls_nonlinear!, w0, tspan, p)

    return x, prob
end;

Reference Solution

Using an adaptive timestepping method to solve the system of ordinary differential equations with high precision.

L = 16.0 # Domain half-length
n = 256 # Number of grid points
alpha = 5.0 # Dispersive coefficient
xs, prob = nonlinear_schrodinger(n, L, alpha)

@time sol = solve(prob, AutoVern7(Rodas5P(autodiff=AutoFiniteDiff()));
                  dt = 1e-4, reltol = 1e-12, abstol = 1e-12);

test_sol = TestSolution(sol) # Reference solution for error estimation

tslices = LinRange(prob.tspan..., 50)
ys_u = mapreduce(t -> sol(t)[1:n], hcat, tslices)
plt = heatmap(xs, tslices, ys_u', xlabel = "x", ylabel = "t", title="Re(ψ)")

27.904516 seconds (9.45 M allocations: 680.597 MiB, 1.02% gc time, 29.63% 
compilation time)

Work-Precision Diagrams

Non-Stiff Explicit Methods

Unlike diffusive PDEs (Burgers, Allen-Cahn, Kuramoto-Sivashinsky), the NLS is dispersive with purely imaginary eigenvalues. Explicit adaptive methods can take large steps on smooth initial conditions without instability, since errors manifest as phase drift rather than exponential blowup.

abstols = 0.1 .^ (5:8)
reltols = 0.1 .^ (2:5)
setups = [
    Dict(:alg => Tsit5()),
    Dict(:alg => Vern7()),
    Dict(:alg => Vern9()),
    Dict(:alg => DP8()),
]
labels = hcat(
    "Tsit5",
    "Vern7",
    "Vern9",
    "DP8",
)
@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(1e7));

plot(wp, label=labels, markershape=:auto, title="Explicit Methods")

Tsit5
Vern7
Vern9
DP8
 72.237706 seconds (22.33 M allocations: 1.610 GiB, 0.97% gc time, 27.51% c
ompilation time)

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-3 * multipliers),
    Dict(:alg => CNAB2(), :dts => 1e-3 * multipliers),
    Dict(:alg => CNLF2(), :dts => 1e-3 * multipliers),
    Dict(:alg => SBDF2(), :dts => 1e-3 * 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="IMEX Methods, High Tolerance")

IMEXEuler
CNAB2
CNLF2
SBDF2
142.779732 seconds (14.23 M allocations: 1.727 GiB, 0.39% gc time, 8.41% co
mpilation 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(krylov=true, m=5), :dts => 1e-3 * multipliers),
    Dict(:alg => NorsettEuler(krylov=true, m=20), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK2(krylov=true, m=5), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-3 * multipliers),
]
labels = hcat(
    "NorsettEuler (m=5)",
    "NorsettEuler (m=20)",
    "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 (m=5)
NorsettEuler (m=20)
ETDRK2 (m=5)
ETDRK2 (m=20)
208.195406 seconds (18.34 M allocations: 1.802 GiB, 0.35% gc time, 6.45% co
mpilation time)

Comparisons Between Families

abstols = 0.1 .^ (5:8)
reltols = 0.1 .^ (1:4)
multipliers = 0.5 .^ (0:3)
setups = [
    Dict(:alg => Vern7()),
    Dict(:alg => CNAB2(), :dts => 1e-3 * multipliers),
    Dict(:alg => CNAB2(linsolve=KrylovJL_GMRES()), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK2(krylov=true, m=20), :dts => 1e-3 * multipliers),
]
labels = hcat(
    "Vern7",
    "CNAB2 (dense linsolve)",
    "CNAB2 (Krylov linsolve)",
    "ETDRK2 (Krylov, 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(1e7));

plot(wp, label=labels, markershape=:auto, title="Between Families, High Tolerances")

Vern7
CNAB2 (dense linsolve)
CNAB2 (Krylov linsolve)
ETDRK2 (Krylov, m=20)
166.016136 seconds (4.34 M allocations: 1.303 GiB, 0.14% gc time, 1.94% com
pilation time)

Low Tolerances

Non-Stiff vs IMEX

Note: KenCarp3/KenCarp4/KenCarp5 were tested but stall at several seconds per solve due to expensive dense linear solves on the $(2N) \times (2N)$ system. They are excluded. ARKODE with dense linear solver is used as the adaptive IMEX representative.

abstols = 0.1 .^ (8:11)
reltols = 0.1 .^ (5:8)
setups = [
    Dict(:alg => Vern7()),
    Dict(:alg => Vern9()),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Dense)),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Dense)),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Dense)),
]
labels = hcat(
    "Vern7",
    "Vern9",
    "ARKODE3",
    "ARKODE4",
    "ARKODE5",
)
@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(1e7));

plot(wp, label=labels, markershape=:auto, title="Non-Stiff vs IMEX, Low Tolerances")

Vern7
Vern9
ARKODE3
ARKODE4
ARKODE5
189.907726 seconds (8.65 M allocations: 427.410 MiB, 0.05% gc time, 1.13% c
ompilation 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(krylov=true, m=20), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK4(krylov=true, m=20), :dts => 1e-3 * multipliers),
    Dict(:alg => HochOst4(krylov=true, m=20), :dts => 1e-3 * multipliers),
]
labels = hcat(
    "ETDRK3 (m=20)",
    "ETDRK4 (m=20)",
    "HochOst4 (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, Low Tolerances")

ETDRK3 (m=20)
ETDRK4 (m=20)
HochOst4 (m=20)
1153.591966 seconds (22.86 M allocations: 7.444 GiB, 0.17% gc time, 0.87% c
ompilation time)

Comparisons Between Families

abstols = 0.1 .^ (7:11)
reltols = 0.1 .^ (4:8)
multipliers = 0.5 .^ (0:4)
setups = [
    Dict(:alg => Vern7()),
    Dict(:alg => Vern9()),
    Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Dense)),
    Dict(:alg => ETDRK3(krylov=true, m=20), :dts => 1e-3 * multipliers),
    Dict(:alg => ETDRK4(krylov=true, m=20), :dts => 1e-3 * multipliers),
]
labels = hcat(
    "Vern7",
    "Vern9",
    "ARKODE5",
    "ETDRK3 (m=20)",
    "ETDRK4 (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(1e7));

plot(wp, label=labels, markershape=:auto, title="Between Families, Low Tolerances")

Vern7
Vern9
ARKODE5
ETDRK3 (m=20)
ETDRK4 (m=20)
778.814474 seconds (5.57 M allocations: 3.627 GiB, 0.07% gc 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","nls_spectral_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`
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⌅ [d4b830b4] OrdinaryDiffEqMultirate v2.7.0
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  [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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  [358294b1] OrdinaryDiffEqStabilizedRK v2.6.0
⌃ [b1df2697] OrdinaryDiffEqTsit5 v2.1.3
⌃ [79d7bb75] OrdinaryDiffEqVerner v2.4.0
  [90014a1f] PDMats v0.11.41
⌅ [d96e819e] Parameters v0.12.3
⌅ [69de0a69] Parsers v2.8.7
  [ccf2f8ad] PlotThemes v3.3.0
  [995b91a9] PlotUtils v1.4.4
  [91a5bcdd] Plots v1.41.7
  [e409e4f3] PoissonRandom v0.4.13
  [f517fe37] Polyester v0.7.19
  [1d0040c9] PolyesterWeave v0.2.2
  [c74db56a] PolynomialBases v0.4.28
⌃ [f27b6e38] Polynomials v4.1.1
⌃ [d236fae5] PreallocationTools v1.6.0
⌅ [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
  [c4ea9172] QuasiArrays v0.13.10
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [b889d2dc] RecurrenceRelationshipArrays v0.1.4
  [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
  [ae5879a3] ResettableStacks v1.4.0
  [9fe22ead] RespecializeParams v1.3.0
  [79098fc4] Rmath v0.9.0
  [47965b36] RootedTrees v2.27.0
⌃ [f2b01f46] Roots v3.0.7
  [7e49a35a] RuntimeGeneratedFunctions v0.5.25
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.46
⌃ [0bca4576] SciMLBase v3.49.2
⌃ [31c91b34] SciMLBenchmarks v0.1.3
⌃ [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`