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Allen-Cahn Pseudospectral Methods Work-Precision Diagrams

Problem Description

The Allen-Cahn partial differential equation is solved on the domain $[-L, L] \times [0, T] \in \mathbb R \times R,~L = 1,~T = 1$, with the following initial and boundary conditions:

\[\begin{aligned} \partial_t u(t,x) & = u(t,x) - u^3(t,x) + \epsilon \partial_x^2 u(t,x), \\ u(0,x) & = \cos(2\pi x), \\ u(t,-L) & = u(t,L) = 1. \end{aligned}\]

The spatial derivative operators are represented via Chebyshev pseudospectral approximations. Here, the domain is discretized by projecting on a cosine-spaced grid of points $x_s \in [-1, 1]$; the solution is approximated on this grid via linear combinations of Chebyshev polynomial basis functions in space. The coefficient $p = 3,~\epsilon = 10^{-3}$ is chosen to produce `interesting' behavior as seen in the reference solution below.

\[\begin{aligned} \frac{du}{dt} & = p(u(t,x) - u^3(t,x)) + \epsilon D_x^2 u(t,x), \\ u(0,x_s) & = \cos(2\pi x_s), \\ u(t,-L) & = u(t,L) = 1. \end{aligned}\]

Implementation

using OrdinaryDiffEq
using OrdinaryDiffEqBDF, OrdinaryDiffEqExponentialRK, OrdinaryDiffEqIMEXMultistep, OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK
using DiffEqDevTools
using SciMLOperators
using LinearSolve
using LinearAlgebra
using SparseArrays
using Sundials
using ClassicalOrthogonalPolynomials
using Plots
gr();
# Nonlinear component of vector field
function f_nonlinear!(du, u, p, t)
    M, D0, alpha, tmp = p
    Tu = D0 * u # Pseudo-spectral representation of solution
    @. tmp[2:end-1] = alpha * (Tu - Tu^3) 
    ldiv!(du, M, tmp) # Solve the linear system M * du = tmp
end

function allen_cahn(n, eps)
    T = ChebyshevT() # Chebyshev basis
    x1 = reverse(ChebyshevGrid{1}(n - 2)) # 1st kind points, sorted
    x2 = reverse(ChebyshevGrid{2}(n)) # 2nd kind points, sorted

    V = T[x2, 1:n] # Vandermonde matrix, its inverse is transform from values to coefficients
    D0 = diff(T, 0)[x1, 1:n] / V # discretisation of identity matrix
    D2 = diff(T, 2)[x1, 1:n] / V # 2nd derivative from x2 to x1

    B_l = [1; zeros(n-1)]'  # Left Dirichlet conditions
    B_r = [zeros(n-1); 1]'  # Right Dirichlet
    M = lu([B_l; D0; B_r]) # Mass matrix for the problem
    u0 = cos.(2π * x2)  # Initial condition
    alpha = 3.0 # Time scaling factor
    D2_bc = [zeros(1, n); D2; zeros(1, n)] # 2nd derivative with Dirichlet conditions

    # Problem setup
    prob = SplitODEProblem(
        MatrixOperator(eps * (M \ D2_bc)), # Linear operator with mass matrix inversion
        f_nonlinear!,
        u0, (0.0, 1.0),
        (M, D0, alpha, similar(u0))
    )

    return x2, prob
end;

Reference Solution

N = 128
eps = 1e-3
xs, prob = allen_cahn(N, eps);

@time sol = solve(prob, Rodas5(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 = mapreduce(sol, hcat, tslices)
heatmap(xs, tslices, ys', xlabel="x", ylabel="t")
8.113695 seconds (6.08 M allocations: 481.439 MiB, 3.38% gc time, 87.57% 
compilation 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
901.290630 seconds (23.88 M allocations: 3.419 GiB, 0.26% gc time, 1.88% co
mpilation time: <1% of which was recompilation)

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)
321.538125 seconds (38.32 M allocations: 13.478 GiB, 1.12% gc time, 4.99% c
ompilation 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 linsolve)",
    "CNAB2 (Krylov linsolve)", 
    "ETDRK2 (m=5)",
)
@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 linsolve)
CNAB2 (Krylov linsolve)
ETDRK2 (m=5)
283.279218 seconds (9.93 M allocations: 6.005 GiB, 1.03% gc time, 1.18% com
pilation time)

Low Tolerances

Implicit-Explicit Methods

Dense/banded linear solvers.

abstols = 0.1 .^ (7:13)
reltols = 0.1 .^ (4:10)
setups = [
    Dict(:alg => KenCarp3()),
    Dict(:alg => KenCarp4()),
    Dict(:alg => KenCarp5()),
    #Dict(:alg => ARKODE(Sundials.Implicit(), order=3, linear_solver=:Band, jac_upper=1, jac_lower=1)),
    #Dict(:alg => ARKODE(Sundials.Implicit(), order=4, linear_solver=:Band, jac_upper=1, jac_lower=1)),
    #Dict(:alg => ARKODE(Sundials.Implicit(), order=5, linear_solver=:Band, jac_upper=1, jac_lower=1))
]
labels = hcat(
    "KenCarp3",
    "KenCarp4",
    "KenCarp5",
    #"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(1e5));

plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Band Linsolve, Low Tolerances")
KenCarp3
KenCarp4
KenCarp5
 54.285220 seconds (32.56 M allocations: 2.131 GiB, 2.29% gc time, 76.26% c
ompilation time)

Krylov linear solvers.

abstols = 0.1 .^ (7:13)
reltols = 0.1 .^ (4:10)
setups = [
    Dict(:alg => KenCarp3(linsolve=KrylovJL_GMRES())),
    Dict(:alg => KenCarp4(linsolve=KrylovJL_GMRES())),
    Dict(:alg => KenCarp5(linsolve=KrylovJL_GMRES())),
    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",
    "KenCarp4",
    "KenCarp5",
    "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(1e5));

plot(wp, label=labels, markershape=:auto, title="IMEX Methods, Krylov Linsolve, Low Tolerances")
KenCarp3
KenCarp4
KenCarp5
ARKODE3
ARKODE4
ARKODE5
 98.753328 seconds (63.88 M allocations: 3.626 GiB, 1.73% gc time, 40.44% 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(), :dts => 1e-2 * multipliers),
    Dict(:alg => ETDRK4(), :dts => 1e-2 * multipliers),
    Dict(:alg => HochOst4(), :dts => 1e-2 * 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)
 20.181402 seconds (10.73 M allocations: 1.855 GiB, 2.29% gc time, 45.33% 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("ARKODE (Krylov linsolve)", "ETDRK3 ()", "ETDRK4 ()")
@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")
ARKODE (Krylov linsolve)
ETDRK3 ()
ETDRK4 ()
 10.049535 seconds (3.77 M allocations: 893.827 MiB, 2.06% gc time, 0.12% 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","allen_cahn_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`
  [47edcb42] ADTypes v1.24.0
  [2169fc97] AlgebraicMultigrid v2.0.2
  [b30e2e7b] ClassicalOrthogonalPolynomials v0.15.20
  [f3b72e0c] DiffEqDevTools v3.6.3
  [40713840] IncompleteLU v0.2.1
  [7f56f5a3] LSODA v1.2.0
⌃ [7ed4a6bd] LinearSolve v5.17.3
  [1dea7af3] OrdinaryDiffEq v7.8.1
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.9
⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.2
  [e0540318] OrdinaryDiffEqExponentialRK v2.4.0
  [5960d6e9] OrdinaryDiffEqFIRK v2.8.7
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v2.2.1
  [9f002381] OrdinaryDiffEqIMEXMultistep v2.3.0
⌅ [d4b830b4] OrdinaryDiffEqMultirate v2.7.0
  [43230ef6] OrdinaryDiffEqRosenbrock v2.7.3
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.4
  [358294b1] OrdinaryDiffEqStabilizedRK v2.7.0
  [91a5bcdd] Plots v1.41.7
  [31c91b34] SciMLBenchmarks v0.2.1
⌃ [c0aeaf25] SciMLOperators v1.30.0
  [9f842d2f] SparseConnectivityTracer v1.2.3
  [0a514795] SparseMatrixColorings v0.4.28
  [9f78cca6] SummationByPartsOperators v0.5.97
  [c3572dad] Sundials v6.7.1
  [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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⌅ [d4b830b4] OrdinaryDiffEqMultirate v2.7.0
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⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.4
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  [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 v4.1.3
  [d236fae5] PreallocationTools v1.7.1
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.6.0
  [08abe8d2] PrettyTables v3.4.8
  [43287f4e] PtrArrays v1.4.0
  [78ab2635] PureGebal v1.1.0
  [0c0d3e7f] PureKLU v1.5.0
  [1fd47b50] QuadGK v2.11.3
  [c4ea9172] QuasiArrays v0.13.11
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [b889d2dc] RecurrenceRelationshipArrays v0.1.5
  [807425ed] RecurrenceRelationships v0.2.0
  [731186ca] RecursiveArrayTools v4.5.1
  [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 v3.0.8
  [7e49a35a] RuntimeGeneratedFunctions v0.5.26
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.46
  [0bca4576] SciMLBase v3.54.0
  [31c91b34] SciMLBenchmarks v0.2.1
  [19f34311] SciMLJacobianOperators v0.1.19
  [a6db7da4] SciMLLogging v2.1.0
⌃ [c0aeaf25] SciMLOperators v1.30.0
  [431bcebd] SciMLPublic v1.3.0
  [53ae85a6] SciMLStructures v1.10.5
  [6c6a2e73] Scratch v1.3.0
  [f8ebbe35] SemiseparableMatrices v0.4.1
  [efcf1570] Setfield v1.1.2
  [992d4aef] Showoff v1.1.1
  [727e6d20] SimpleNonlinearSolve v2.14.5
  [ce78b400] SimpleUnPack v1.1.0
  [a2af1166] SortingAlgorithms v1.2.3
  [a57abbd0] SparseColumnPivotedQR v2.1.8
  [9f842d2f] SparseConnectivityTracer v1.2.3
  [0a514795] SparseMatrixColorings v0.4.28
  [276daf66] SpecialFunctions v2.9.0
  [860ef19b] StableRNGs v1.0.4
  [aedffcd0] Static v1.4.6
  [0d7ed370] StaticArrayInterface v1.10.0
  [90137ffa] StaticArrays v1.9.20
  [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
⌅ [892a3eda] StringManipulation v0.5.0
  [09ab397b] StructArrays v0.7.3
  [9f78cca6] SummationByPartsOperators v0.5.97
  [c3572dad] Sundials v6.7.1
  [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 v1.2.1
  [c751599d] ToeplitzMatrices v0.8.5
  [781d530d] TruncatedStacktraces v1.4.0
  [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
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.16
  [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 v2.8.4+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 v3.5.1+0
  [d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [7746bdde] Glib_jll v2.88.3+0
  [3b182d85] Graphite2_jll v1.3.16+0
  [2e76f6c2] HarfBuzz_jll v100.14004.0+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 v4.2.0+0
  [1d63c593] LLVMOpenMP_jll v23.1.1+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
  [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
⌅ [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 v5.8.4+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 v1.1.1
  [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 v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [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 v17.4.0+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`