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Units don't work correctly #511

Description

@ctessum

Describe the bug 🐞

A duplicate of #287 , but with an updated example

Expected behavior

I would expect discretization to work correctly when units are involved.

Minimal Reproducible Example 👇

using ModelingToolkit
using ModelingToolkit: t, D 
using MethodOfLines, DomainSets, DynamicQuantities, OrdinaryDiffEqDefault

@parameters x [unit = u"m"]
@parameters D_diff [description = "Diffusion coefficient", unit = u"m^2/s"]
@variables u(..) [unit = u"kg/m^3"]

@constants begin
    u_ref = 1.0, [description = "Reference concentration", unit = u"kg/m^3"]
    x_ref = 1.0, [description = "Reference length", unit = u"m"]
end

Dx = Differential(x)
Dxx = Dx  Dx  # Second derivative operator

eq = [D(u(t, x)) ~ D_diff * Dxx(u(t, x))]  # Continuous PDE form

# Define spatial domain and boundary/initial conditions
domains = [t  Interval(0.0, 1.0),
           x  Interval(0.0, 1.0)]

bcs = [u(0, x) ~ u_ref * sin* x / x_ref),  # Initial condition
       u(t, 0) ~ 0 * u_ref,                   # Left boundary
       u(t, 1) ~ 0 * u_ref]                   # Right boundary

@named sys = PDESystem(eq, bcs, domains, [t, x], [u(t, x)], [D_diff => 0.1])

dx = 0.05
discretization = MOLFiniteDifference([x => dx], t)
prob = MethodOfLines.discretize(sys, discretization)
sol = solve(prob)

Error & Stacktrace ⚠️

┌ Warning:  in eq. #1left, in sum Differential(t)((u(t))[2]) - D_diff*(399.99999999999994(u(t))[1] - 799.9999999999999(u(t))[2] + 399.99999999999994(u(t))[3]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #2left, in sum Differential(t)((u(t))[3]) - D_diff*(399.99999999999994(u(t))[2] - 799.9999999999999(u(t))[3] + 399.99999999999994(u(t))[4]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #3left, in sum Differential(t)((u(t))[4]) - D_diff*(399.99999999999994(u(t))[3] - 799.9999999999999(u(t))[4] + 399.99999999999994(u(t))[5]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #4left, in sum Differential(t)((u(t))[5]) - D_diff*(399.99999999999994(u(t))[4] - 799.9999999999999(u(t))[5] + 399.99999999999994(u(t))[6]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #5left, in sum Differential(t)((u(t))[6]) - D_diff*(399.99999999999994(u(t))[5] - 799.9999999999999(u(t))[6] + 399.99999999999994(u(t))[7]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #6left, in sum Differential(t)((u(t))[7]) - D_diff*(399.99999999999994(u(t))[6] - 799.9999999999999(u(t))[7] + 399.99999999999994(u(t))[8]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #7left, in sum Differential(t)((u(t))[8]) - D_diff*(399.99999999999994(u(t))[7] - 799.9999999999999(u(t))[8] + 399.99999999999994(u(t))[9]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #8left, in sum Differential(t)((u(t))[9]) - D_diff*(399.99999999999994(u(t))[10] + 399.99999999999994(u(t))[8] - 799.9999999999999(u(t))[9]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #9left, in sum Differential(t)((u(t))[10]) - D_diff*(-799.9999999999999(u(t))[10] + 399.99999999999994(u(t))[11] + 399.99999999999994(u(t))[9]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #10left, in sum Differential(t)((u(t))[11]) - D_diff*(399.99999999999994(u(t))[10] - 799.9999999999999(u(t))[11] + 399.99999999999994(u(t))[12]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #11left, in sum Differential(t)((u(t))[12]) - D_diff*(399.99999999999994(u(t))[11] - 799.9999999999999(u(t))[12] + 399.99999999999994(u(t))[13]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #12left, in sum Differential(t)((u(t))[13]) - D_diff*(399.99999999999994(u(t))[12] - 799.9999999999999(u(t))[13] + 399.99999999999994(u(t))[14]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #13left, in sum Differential(t)((u(t))[14]) - D_diff*(399.99999999999994(u(t))[13] - 799.9999999999999(u(t))[14] + 399.99999999999994(u(t))[15]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #14left, in sum Differential(t)((u(t))[15]) - D_diff*(399.99999999999994(u(t))[14] - 799.9999999999999(u(t))[15] + 399.99999999999994(u(t))[16]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #15left, in sum Differential(t)((u(t))[16]) - D_diff*(399.99999999999994(u(t))[15] - 799.9999999999999(u(t))[16] + 399.99999999999994(u(t))[17]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #16left, in sum Differential(t)((u(t))[17]) - D_diff*(399.99999999999994(u(t))[16] - 799.9999999999999(u(t))[17] + 399.99999999999994(u(t))[18]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #17left, in sum Differential(t)((u(t))[18]) - D_diff*(399.99999999999994(u(t))[17] - 799.9999999999999(u(t))[18] + 399.99999999999994(u(t))[19]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #18left, in sum Differential(t)((u(t))[19]) - D_diff*(399.99999999999994(u(t))[18] - 799.9999999999999(u(t))[19] + 399.99999999999994(u(t))[20]), units [1.0 m² s⁻¹, 1.0 s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
┌ Warning:  in eq. #19left, in sum Differential(t)((u(t))[20]) - D_diff*(399.99999999999994(u(t))[19] - 799.9999999999999(u(t))[20] + 399.99999999999994(u(t))[21]), units [1.0 s⁻¹, 1.0 m² s⁻¹] do not match.
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:181
The system of equations is:

Discretization failed, please post an issue on https://github.com/SciML/MethodOfLines.jl with the failing code and system at low point count.

ERROR: ModelingToolkit.ValidationError("Some equations had invalid units. See warnings for details.")
Stacktrace:
 [1] check_units
   @ ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/unit_check.jl:329 [inlined]
 [2] System(tag::UInt64, eqs::Vector{…}, noise_eqs::Nothing, jumps::Vector{…}, constraints::Vector{…}, costs::Vector{…}, consolidate::typeof(ModelingToolkit.default_consolidate), unknowns::Vector{…}, ps::Vector{…}, brownians::Vector{…}, iv::SymbolicUtils.BasicSymbolic{…}, observed::Vector{…}, parameter_dependencies::Vector{…}, var_to_name::Dict{…}, name::Symbol, description::String, defaults::Dict{…}, guesses::Dict{…}, systems::Vector{…}, initialization_eqs::Vector{…}, continuous_events::Vector{…}, discrete_events::Vector{…}, connector_type::Nothing, assertions::Dict{…}, metadata::Base.ImmutableDict{…}, gui_metadata::Nothing, is_dde::Bool, tstops::Vector{…}, inputs::OrderedCollections.OrderedSet{…}, outputs::OrderedCollections.OrderedSet{…}, tearing_state::Nothing, namespacing::Bool, complete::Bool, index_cache::Nothing, ignored_connections::Nothing, preface::Vector{…}, parent::Nothing, initializesystem::Nothing, is_initializesystem::Bool, is_discrete::Bool, isscheduled::Bool, schedule::Nothing; checks::Bool)
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/system.jl:296
 [3] System
   @ ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/system.jl:263 [inlined]
 [4] System(eqs::Vector{…}, iv::Num, dvs::Vector{…}, ps::Vector{…}, brownians::Vector{…}; constraints::Vector{…}, noise_eqs::Nothing, jumps::Vector{…}, costs::Vector{…}, consolidate::Function, observed::Vector{…}, parameter_dependencies::Vector{…}, defaults::Dict{…}, guesses::Dict{…}, systems::Vector{…}, initialization_eqs::Vector{…}, continuous_events::Vector{…}, discrete_events::Vector{…}, connector_type::Nothing, assertions::Dict{…}, metadata::Vector{…}, gui_metadata::Nothing, is_dde::Nothing, tstops::Vector{…}, inputs::OrderedCollections.OrderedSet{…}, outputs::OrderedCollections.OrderedSet{…}, tearing_state::Nothing, ignored_connections::Nothing, parent::Nothing, description::String, name::Symbol, discover_from_metadata::Bool, initializesystem::Nothing, is_initializesystem::Bool, is_discrete::Bool, preface::Vector{…}, checks::Bool)
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/JvjlW/src/systems/system.jl:452
 [5] generate_system(disc_state::PDEBase.EquationState, s::MethodOfLines.DiscreteSpace{…}, u0::Vector{…}, tspan::Tuple{…}, metadata::MethodOfLines.MOLMetadata{…}, disc::MOLFiniteDifference{…})
   @ PDEBase ~/.julia/packages/PDEBase/CpWUp/src/discretization_state.jl:40
 [6] symbolic_discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…})
   @ PDEBase ~/.julia/packages/PDEBase/CpWUp/src/symbolic_discretize.jl:90
 [7] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…}; analytic::Nothing, kwargs::@Kwargs{})
   @ PDEBase ~/.julia/packages/PDEBase/CpWUp/src/discretization_state.jl:57
 [8] discretize(pdesys::PDESystem, discretization::MOLFiniteDifference{…})
   @ PDEBase ~/.julia/packages/PDEBase/CpWUp/src/discretization_state.jl:54
 [9] top-level scope
   @...
Some type information was truncated. Use `show(err)` to see complete types.

Environment (please complete the following information):

  • Output of using Pkg; Pkg.status()
  [5b8099bc] DomainSets v0.7.16
  [06fc5a27] DynamicQuantities v1.11.0
  [94925ecb] MethodOfLines v0.11.9
⌅ [961ee093] ModelingToolkit v10.32.1
  [50262376] OrdinaryDiffEqDefault v1.12.0
  • Output of using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)
  [47edcb42] ADTypes v1.21.0
  [1520ce14] AbstractTrees v0.4.5
  [7d9f7c33] Accessors v0.1.43
  [79e6a3ab] Adapt v4.4.0
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
  [4fba245c] ArrayInterface v7.22.0
  [4c555306] ArrayLayouts v1.12.2
  [13072b0f] AxisAlgorithms v1.1.0
  [e2ed5e7c] Bijections v0.2.2
  [62783981] BitTwiddlingConvenienceFunctions v0.1.6
  [8e7c35d0] BlockArrays v1.9.3
  [70df07ce] BracketingNonlinearSolve v1.6.2
  [2a0fbf3d] CPUSummary v0.2.7
  [d360d2e6] ChainRulesCore v1.26.0
  [fb6a15b2] CloseOpenIntervals v0.1.13
⌅ [861a8166] Combinatorics v1.0.2
  [a80b9123] CommonMark v0.10.3
  [38540f10] CommonSolve v0.2.6
  [bbf7d656] CommonSubexpressions v0.3.1
  [f70d9fcc] CommonWorldInvalidations v1.0.0
  [34da2185] Compat v4.18.1
  [b152e2b5] CompositeTypes v0.1.4
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
  [187b0558] ConstructionBase v1.6.0
  [adafc99b] CpuId v0.3.1
  [9a962f9c] DataAPI v1.16.0
  [864edb3b] DataStructures v0.19.3
  [2b5f629d] DiffEqBase v6.200.0
  [459566f4] DiffEqCallbacks v4.12.0
  [77a26b50] DiffEqNoiseProcess v5.26.0
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
  [a0c0ee7d] DifferentiationInterface v0.7.16
  [8d63f2c5] DispatchDoctor v0.4.28
  [31c24e10] Distributions v0.25.123
  [ffbed154] DocStringExtensions v0.9.5
  [5b8099bc] DomainSets v0.7.16
  [7c1d4256] DynamicPolynomials v0.6.4
  [06fc5a27] DynamicQuantities v1.11.0
  [4e289a0a] EnumX v1.0.6
  [f151be2c] EnzymeCore v0.8.18
  [d4d017d3] ExponentialUtilities v1.30.0
  [e2ba6199] ExprTools v0.1.10
  [55351af7] ExproniconLite v0.10.14
  [7034ab61] FastBroadcast v0.3.5
  [9aa1b823] FastClosures v0.3.2
  [442a2c76] FastGaussQuadrature v1.1.0
  [a4df4552] FastPower v1.3.1
  [1a297f60] FillArrays v1.16.0
  [64ca27bc] FindFirstFunctions v1.8.0
  [6a86dc24] FiniteDiff v2.29.0
  [1fa38f19] Format v1.3.7
⌃ [f6369f11] ForwardDiff v0.10.39
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [46192b85] GPUArraysCore v0.2.0
  [c145ed77] GenericSchur v0.5.6
  [c27321d9] Glob v1.4.0
  [86223c79] Graphs v1.13.4
  [34004b35] HypergeometricFunctions v0.3.28
  [615f187c] IfElse v0.1.1
  [3263718b] ImplicitDiscreteSolve v1.7.0
  [d25df0c9] Inflate v0.1.5
  [18e54dd8] IntegerMathUtils v0.1.3
  [a98d9a8b] Interpolations v0.16.2
  [8197267c] IntervalSets v0.7.13
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.6
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.7.1
  [ae98c720] Jieko v0.2.1
  [98e50ef6] JuliaFormatter v2.3.0
⌅ [70703baa] JuliaSyntax v0.4.10
  [ccbc3e58] JumpProcesses v9.21.2
  [ba0b0d4f] Krylov v0.10.5
  [b964fa9f] LaTeXStrings v1.4.0
  [23fbe1c1] Latexify v0.16.10
  [10f19ff3] LayoutPointers v0.1.17
  [87fe0de2] LineSearch v0.1.6
  [d3d80556] LineSearches v7.6.0
  [7ed4a6bd] LinearSolve v3.58.0
  [2ab3a3ac] LogExpFunctions v0.3.29
  [e6f89c97] LoggingExtras v1.2.0
  [d8e11817] MLStyle v0.4.17
  [1914dd2f] MacroTools v0.5.16
  [d125e4d3] ManualMemory v0.1.8
  [bb5d69b7] MaybeInplace v0.1.4
  [94925ecb] MethodOfLines v0.11.9
  [e1d29d7a] Missings v1.2.0
⌅ [961ee093] ModelingToolkit v10.32.1
  [2e0e35c7] Moshi v0.3.7
  [46d2c3a1] MuladdMacro v0.2.4
  [102ac46a] MultivariatePolynomials v0.5.13
  [d8a4904e] MutableArithmetics v1.6.7
  [d41bc354] NLSolversBase v8.0.0
  [77ba4419] NaNMath v1.1.3
⌃ [8913a72c] NonlinearSolve v4.12.0
  [be0214bd] NonlinearSolveBase v2.11.2
  [5959db7a] NonlinearSolveFirstOrder v1.11.1
  [9a2c21bd] NonlinearSolveQuasiNewton v1.12.0
  [26075421] NonlinearSolveSpectralMethods v1.6.0
  [6fe1bfb0] OffsetArrays v1.17.0
  [bac558e1] OrderedCollections v1.8.1
  [1dea7af3] OrdinaryDiffEq v6.108.0
  [89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.9.0
  [6ad6398a] OrdinaryDiffEqBDF v1.16.0
  [bbf590c4] OrdinaryDiffEqCore v3.4.2
  [50262376] OrdinaryDiffEqDefault v1.12.0
  [4302a76b] OrdinaryDiffEqDifferentiation v2.0.0
  [9286f039] OrdinaryDiffEqExplicitRK v1.8.0
  [e0540318] OrdinaryDiffEqExponentialRK v1.13.0
  [becaefa8] OrdinaryDiffEqExtrapolation v1.15.0
  [5960d6e9] OrdinaryDiffEqFIRK v1.22.0
  [101fe9f7] OrdinaryDiffEqFeagin v1.8.0
  [d3585ca7] OrdinaryDiffEqFunctionMap v1.9.0
  [d28bc4f8] OrdinaryDiffEqHighOrderRK v1.9.0
  [9f002381] OrdinaryDiffEqIMEXMultistep v1.12.0
  [521117fe] OrdinaryDiffEqLinear v1.10.0
  [1344f307] OrdinaryDiffEqLowOrderRK v1.10.0
  [b0944070] OrdinaryDiffEqLowStorageRK v1.12.0
  [127b3ac7] OrdinaryDiffEqNonlinearSolve v1.20.0
  [c9986a66] OrdinaryDiffEqNordsieck v1.9.0
  [5dd0a6cf] OrdinaryDiffEqPDIRK v1.11.0
  [5b33eab2] OrdinaryDiffEqPRK v1.8.0
  [04162be5] OrdinaryDiffEqQPRK v1.8.0
  [af6ede74] OrdinaryDiffEqRKN v1.9.0
  [43230ef6] OrdinaryDiffEqRosenbrock v1.23.0
  [2d112036] OrdinaryDiffEqSDIRK v1.12.0
  [669c94d9] OrdinaryDiffEqSSPRK v1.11.0
  [e3e12d00] OrdinaryDiffEqStabilizedIRK v1.11.0
  [358294b1] OrdinaryDiffEqStabilizedRK v1.8.0
  [fa646aed] OrdinaryDiffEqSymplecticRK v1.11.0
  [b1df2697] OrdinaryDiffEqTsit5 v1.9.0
  [79d7bb75] OrdinaryDiffEqVerner v1.11.0
⌃ [a7812802] PDEBase v0.1.19
  [90014a1f] PDMats v0.11.37
  [e409e4f3] PoissonRandom v0.4.7
  [f517fe37] Polyester v0.7.18
  [1d0040c9] PolyesterWeave v0.2.2
⌃ [d236fae5] PreallocationTools v0.4.34
  [aea7be01] PrecompileTools v1.3.3
  [21216c6a] Preferences v1.5.1
  [27ebfcd6] Primes v0.5.7
  [43287f4e] PtrArrays v1.3.0
  [1fd47b50] QuadGK v2.11.2
  [74087812] Random123 v1.7.1
  [e6cf234a] RandomNumbers v1.6.0
  [c84ed2f1] Ratios v0.4.5
  [3cdcf5f2] RecipesBase v1.3.4
  [731186ca] RecursiveArrayTools v3.47.0
  [189a3867] Reexport v1.2.2
  [ae029012] Requires v1.3.1
  [ae5879a3] ResettableStacks v1.2.0
  [79098fc4] Rmath v0.9.0
  [7e49a35a] RuntimeGeneratedFunctions v0.5.17
  [9dfe8606] SCCNonlinearSolve v1.9.0
  [94e857df] SIMDTypes v0.1.0
  [0bca4576] SciMLBase v2.138.0
  [19f34311] SciMLJacobianOperators v0.1.12
  [a6db7da4] SciMLLogging v1.8.0
  [c0aeaf25] SciMLOperators v1.15.1
  [431bcebd] SciMLPublic v1.0.1
  [53ae85a6] SciMLStructures v1.10.0
  [efcf1570] Setfield v1.1.2
  [727e6d20] SimpleNonlinearSolve v2.10.0
  [699a6c99] SimpleTraits v0.9.5
  [a2af1166] SortingAlgorithms v1.2.2
  [0a514795] SparseMatrixColorings v0.4.23
  [276daf66] SpecialFunctions v2.6.1
  [aedffcd0] Static v1.3.1
  [0d7ed370] StaticArrayInterface v1.8.0
  [90137ffa] StaticArrays v1.9.16
  [1e83bf80] StaticArraysCore v1.4.4
  [10745b16] Statistics v1.11.1
  [82ae8749] StatsAPI v1.8.0
  [2913bbd2] StatsBase v0.34.10
  [4c63d2b9] StatsFuns v1.5.2
  [7792a7ef] StrideArraysCore v0.5.8
  [2efcf032] SymbolicIndexingInterface v0.3.46
⌅ [19f23fe9] SymbolicLimits v0.2.3
⌅ [d1185830] SymbolicUtils v3.32.0
⌅ [0c5d862f] Symbolics v6.58.0
  [ed4db957] TaskLocalValues v0.1.3
  [8ea1fca8] TermInterface v2.0.0
  [1c621080] TestItems v1.0.0
  [8290d209] ThreadingUtilities v0.5.5
  [a759f4b9] TimerOutputs v0.5.29
  [410a4b4d] Tricks v0.1.13
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.6.1
  [3a884ed6] UnPack v1.0.2
  [1986cc42] Unitful v1.28.0
  [a7c27f48] Unityper v0.1.6
  [efce3f68] WoodburyMatrices v1.1.0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [aacddb02] JpegTurbo_jll v3.1.4+0
  [856f044c] MKL_jll v2025.2.0+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [f50d1b31] Rmath_jll v0.5.1+0
  [1317d2d5] oneTBB_jll v2022.0.0+1
  [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 v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf 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 v1.12.0
  [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 v1.3.0+1
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.4+0
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+0
  • Output of versioninfo()
Julia Version 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 8 × Apple M1
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, apple-m1)
  GC: Built with stock GC
Threads: 4 default, 1 interactive, 4 GC (on 4 virtual cores)
Environment:
  JULIA_EDITOR = code
  JULIA_VSCODE_REPL = 1
  JULIA_NUM_THREADS = 4

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