For solving a two-point boundary value problem using SciML FIRK solvers (RadauIIa5, in this case) I find that the solution process does not treat the initial guess values properly, either by using a vector initial guess or a previously calculated solution object. As shown in the accompanying MWE output, the boundary condition function at t=0 (function bca! in the MWE shown here) is first called, and it uses the proper values of the state variables from the initial guess. Then the function bcb!, at t=1, is called for the first time and it uses the state variables’ guessed values at t=0, that is, the values for the wrong boundary condition. It takes the 6th call to bcb! for the proper values for t=1 to appear.
This issue makes solution of a complex, nonlinear two-point BVP by the continuation method impractical, since no matter how close the guess is to the converged solution, the use of incorrect starting values in the t=1 boundary condition function may give potentially exception-producing results.
Minimal Reproducible Example 👇
Description of the MWE: A simple, linear, thermal problem of a bar of material extending from t=0 to t=1, with uniform internal heat generation. At t=0 the temperature is fixed at 100 degrees C and the axial heat flow is zero. At t=1 the heat flow is convective with an unknown coefficient of convection and an environmental temperature of 20 degrees C. I use a FIRK solver (the same method as in my actual, highly nonlinear problem) to solve for the temperature distribution and the convective coefficient.
In the first call to the RaduaIIa5 solver I use a vector of vectors as the initial guess. These values are from a converged case. In the second call to the solver I use a solution (sol1) object as the initial guess. I print the value part of the dual numbers that represent the three state variables (temperature, heat flux, and convective coefficient) for every time the boundary condition functions are called.
using BoundaryValueDiffEq, NonlinearSolve, ForwardDiff
tspan = (0.0, 1.0)
function f!(du, u, p, t)
cond = 0.002
vol_heat = 0.2
du[1] = -u[2]/cond
du[2] = vol_heat
du[3] = 0.0
end
function bca!(res_a, u_a, p)
res_a[1] = u_a[2]
res_a[2] = u_a[1] - 100.0
if p[1]>0.0
println("t = 0.0 ", " u_a[1] = ", ForwardDiff.value(u_a[1]), " u_a[2] = ", ForwardDiff.value(u_a[2]), " u_a[3] = ", ForwardDiff.value(u_a[3]))
println("")
end
end
function bcb!(res_b, u_b, p)
tref = 20.0
res_b[1] = u_b[3] * (u_b[1] - tref) - u_b[2]
if p[1]>0.0
println("t = 1.0 ", " u_b[1] = ", ForwardDiff.value(u_b[1]), " u_b[2] = ", ForwardDiff.value(u_b[2]), " u_b[3] = ", ForwardDiff.value(u_b[3]))
println("")
end
end
u_guess = [ [100.0, 0.0, 0.006666666666666668],
[99.5, 0.020000000000000004, 0.006666666666666668],
[98.0, 0.04000000000000001, 0.006666666666666668],
[95.5, 0.060000000000000005, 0.006666666666666668],
[92.0, 0.08000000000000002, 0.006666666666666668],
[87.5, 0.1, 0.006666666666666668],
[82.0, 0.12000000000000001, 0.006666666666666668],
[75.5, 0.14, 0.006666666666666668],
[68.0, 0.16000000000000003, 0.006666666666666668],
[59.49999999999999, 0.18000000000000002, 0.006666666666666668],
[50.0, 0.2, 0.006666666666666668] ]
bvp1 = TwoPointBVProblem(f!, (bca!, bcb!), u_guess, tspan, [-1.0]; bcresid_prototype = (zeros(2), zeros(1)))
sol1 = solve(bvp1, RadauIIa5(nlsolve = NewtonRaphson()), dt = 0.1, adaptive = false; nlsolve_kwargs = (; maxiters = 10, show_trace = Val(true)))
bvp2 = TwoPointBVProblem(f!, (bca!, bcb!), sol1, tspan, [1.0]; bcresid_prototype = (zeros(2), zeros(1)))
sol2 = solve(bvp2, RadauIIa5(nlsolve = NewtonRaphson()), dt = 0.1, adaptive = false; nlsolve_kwargs = (; maxiters = 10, show_trace = Val(false)))Error & Output ⚠️
Note the output from the first line beginning with "t = 1.0"
Algorithm: NewtonRaphson(
descent = NewtonDescent(),
autodiff = AutoForwardDiff(),
vjp_autodiff = AutoFiniteDiff(
fdtype = Val{:forward}(),
fdjtype = Val{:forward}(),
fdhtype = Val{:hcentral}(),
dir = true
),
jvp_autodiff = AutoForwardDiff(),
concrete_jac = Val{false}()
)
---- ------------- -----------
Iter f(u) 2-norm Step 2-norm
---- ------------- -----------
0 7.07815150e+02 0.00000000e+00
1 2.49915494e-13 1.08699193e+03
Final 2.49915494e-13
----------------------
t = 0.0 u_a[1] = 100.0 u_a[2] = -7.966160064343519e-17 u_a[3] = 0.0066666666666666706
t = 1.0 u_b[1] = 100.0 u_b[2] = -7.966160064343519e-17 u_b[3] = 0.0066666666666666706
t = 0.0 u_a[1] = 100.0 u_a[2] = -7.966160064343519e-17 u_a[3] = 0.0066666666666666706
t = 1.0 u_b[1] = 100.0 u_b[2] = -7.966160064343519e-17 u_b[3] = 0.0066666666666666706
t = 0.0 u_a[1] = 100.0 u_a[2] = -7.966160064343519e-17 u_a[3] = 0.0066666666666666706
t = 1.0 u_b[1] = 100.0 u_b[2] = -7.966160064343519e-17 u_b[3] = 0.0066666666666666706
t = 0.0 u_a[1] = 100.0 u_a[2] = -7.966160064343519e-17 u_a[3] = 0.0066666666666666706
t = 1.0 u_b[1] = 100.0 u_b[2] = -7.966160064343519e-17 u_b[3] = 0.0066666666666666706
t = 0.0 u_a[1] = 100.0 u_a[2] = -7.966160064343519e-17 u_a[3] = 0.0066666666666666706
t = 1.0 u_b[1] = 100.0 u_b[2] = -7.966160064343519e-17 u_b[3] = 0.0066666666666666706
t = 0.0 u_a[1] = 100.0 u_a[2] = -1.0967901537864265e-16 u_a[3] = 0.006666666666666661
t = 1.0 u_b[1] = 50.000000000000064 u_b[2] = 0.19999999999999982 u_b[3] = 0.006666666666666661
t = 0.0 u_a[1] = 100.0 u_a[2] = -1.0967901537864265e-16 u_a[3] = 0.006666666666666661
t = 1.0 u_b[1] = 50.000000000000064 u_b[2] = 0.19999999999999982 u_b[3] = 0.006666666666666661
t = 0.0 u_a[1] = 100.0 u_a[2] = -1.0967901537864265e-16 u_a[3] = 0.006666666666666661
t = 1.0 u_b[1] = 50.000000000000064 u_b[2] = 0.19999999999999982 u_b[3] = 0.006666666666666661Environment (please complete the following information):
- Output of
using Pkg; Pkg.status()
Status `C:\Users\jbmcv\OneDrive\Documents\MasterPC\New Model\Project.toml`
[764a87c0] BoundaryValueDiffEq v5.20.0
[85d9eb09] BoundaryValueDiffEqFIRK v1.12.0
[336ed68f] CSV v0.10.16
[a93c6f00] DataFrames v1.8.1
[82cc6244] DataInterpolations v8.9.0
[f6369f11] ForwardDiff v1.3.2
[c27321d9] Glob v1.4.0
[033835bb] JLD2 v0.6.3
[87fe0de2] LineSearch v0.1.6
[d3d80556] LineSearches v7.6.0
[77ba4419] NaNMath v1.1.3
[8913a72c] NonlinearSolve v4.16.0
[1dea7af3] OrdinaryDiffEq v6.108.0
[91a5bcdd] Plots v1.41.6
[276daf66] SpecialFunctions v2.7.1
[10745b16] Statistics v1.11.1
[fdbf4ff8] XLSX v0.10.4
[de0858da] Printf v1.11.0
- Output of
using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)
Status `C:\Users\jbmcv\OneDrive\Documents\MasterPC\New Model\Manifest.toml`
[47edcb42] ADTypes v1.21.0
[7d9f7c33] Accessors v0.1.43
[79e6a3ab] Adapt v4.5.0
[66dad0bd] AliasTables v1.1.3
[a95523ee] AlmostBlockDiagonals v0.1.10
[dce04be8] ArgCheck v2.5.0
[4fba245c] ArrayInterface v7.23.0
[4c555306] ArrayLayouts v1.12.2
[aae01518] BandedMatrices v1.11.0
[d1d4a3ce] BitFlags v0.1.9
[62783981] BitTwiddlingConvenienceFunctions v0.1.6
[764a87c0] BoundaryValueDiffEq v5.20.0
[7227322d] BoundaryValueDiffEqAscher v1.11.0
[56b672f2] BoundaryValueDiffEqCore v2.1.0
[85d9eb09] BoundaryValueDiffEqFIRK v1.12.0
[1a22d4ce] BoundaryValueDiffEqMIRK v1.12.0
[9255f1d6] BoundaryValueDiffEqMIRKN v1.11.0
[ed55bfe0] BoundaryValueDiffEqShooting v1.12.0
[70df07ce] BracketingNonlinearSolve v1.11.0
[2a0fbf3d] CPUSummary v0.2.7
[336ed68f] CSV v0.10.16
[d360d2e6] ChainRulesCore v1.26.0
[0b6fb165] ChunkCodecCore v1.0.1
[4c0bbee4] ChunkCodecLibZlib v1.0.0
[55437552] ChunkCodecLibZstd v1.0.0
[fb6a15b2] CloseOpenIntervals v0.1.13
[6309b1aa] CodecInflate64 v0.1.3
[944b1d66] CodecZlib v0.7.8
[35d6a980] ColorSchemes v3.31.0
[3da002f7] ColorTypes v0.12.1
[c3611d14] ColorVectorSpace v0.11.0
[5ae59095] Colors v0.13.1
[861a8166] Combinatorics v1.1.0
[38540f10] CommonSolve v0.2.6
[bbf7d656] CommonSubexpressions v0.3.1
[f70d9fcc] CommonWorldInvalidations v1.0.0
[34da2185] Compat v4.18.1
[a33af91c] CompositionsBase v0.1.2
[2569d6c7] ConcreteStructs v0.2.3
[f0e56b4a] ConcurrentUtilities v2.5.1
[187b0558] ConstructionBase v1.6.0
[d38c429a] Contour v0.6.3
[adafc99b] CpuId v0.3.1
[a8cc5b0e] Crayons v4.1.1
[9a962f9c] DataAPI v1.16.0
[a93c6f00] DataFrames v1.8.1
[82cc6244] DataInterpolations v8.9.0
[864edb3b] DataStructures v0.19.3
[e2d170a0] DataValueInterfaces v1.0.0
[8bb1440f] DelimitedFiles v1.9.1
[2b5f629d] DiffEqBase v6.211.0
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.15.1
[a0c0ee7d] DifferentiationInterface v0.7.16
[31c24e10] Distributions v0.25.123
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[4e289a0a] EnumX v1.0.7
[f151be2c] EnzymeCore v0.8.18
[460bff9d] ExceptionUnwrapping v0.1.11
[d4d017d3] ExponentialUtilities v1.30.0
[e2ba6199] ExprTools v0.1.10
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[9aa1b823] FastClosures v0.3.2
[442a2c76] FastGaussQuadrature v1.1.0
[a4df4552] FastPower v1.3.1
[5789e2e9] FileIO v1.18.0
[48062228] FilePathsBase v0.9.24
[1a297f60] FillArrays v1.16.0
[64ca27bc] FindFirstFunctions v1.8.0
[6a86dc24] FiniteDiff v2.29.0
[53c48c17] FixedPointNumbers v0.8.5
[1fa38f19] Format v1.3.7
[f6369f11] ForwardDiff v1.3.2
[069b7b12] FunctionWrappers v1.1.3
[77dc65aa] FunctionWrappersWrappers v0.1.3
[46192b85] GPUArraysCore v0.2.0
[28b8d3ca] GR v0.73.24
[c145ed77] GenericSchur v0.5.6
[c27321d9] Glob v1.4.0
[42e2da0e] Grisu v1.0.2
[19dc6840] HCubature v1.8.0
[cd3eb016] HTTP v1.11.0
[076d061b] HashArrayMappedTries v0.2.0
[34004b35] HypergeometricFunctions v0.3.28
[615f187c] IfElse v0.1.1
[842dd82b] InlineStrings v1.4.5
[0c81fc1b] InputBuffers v1.1.1
[18e54dd8] IntegerMathUtils v0.1.3
[de52edbc] Integrals v5.1.0
[3587e190] InverseFunctions v0.1.17
[41ab1584] InvertedIndices v1.3.1
[92d709cd] IrrationalConstants v0.2.6
[82899510] IteratorInterfaceExtensions v1.0.0
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[1019f520] JLFzf v0.1.11
[692b3bcd] JLLWrappers v1.7.1
[682c06a0] JSON v1.4.0
[ae98c720] Jieko v0.2.1
[ba0b0d4f] Krylov v0.10.6
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[5078a376] LazyArrays v2.9.5
[87fe0de2] LineSearch v0.1.6
[d3d80556] LineSearches v7.6.0
[7ed4a6bd] LinearSolve v3.66.0
[2ab3a3ac] LogExpFunctions v0.3.29
[e6f89c97] LoggingExtras v1.2.0
[1914dd2f] MacroTools v0.5.16
[d125e4d3] ManualMemory v0.1.8
[a3b82374] MatrixFactorizations v3.1.3
[bb5d69b7] MaybeInplace v0.1.4
[739be429] MbedTLS v1.1.10
[442fdcdd] Measures v0.3.3
[e1d29d7a] Missings v1.2.0
[4886b29c] MonteCarloIntegration v0.2.0
[2e0e35c7] Moshi v0.3.7
[46d2c3a1] MuladdMacro v0.2.4
[d41bc354] NLSolversBase v8.0.0
[77ba4419] NaNMath v1.1.3
[8913a72c] NonlinearSolve v4.16.0
[be0214bd] NonlinearSolveBase v2.15.0
[5959db7a] NonlinearSolveFirstOrder v2.0.0
[9a2c21bd] NonlinearSolveQuasiNewton v1.12.0
[26075421] NonlinearSolveSpectralMethods v1.6.0
[4d8831e6] OpenSSL v1.6.1
[bca83a33] OptimizationBase v5.1.0
[bac558e1] OrderedCollections v1.8.1
[1dea7af3] OrdinaryDiffEq v6.108.0
[89bda076] OrdinaryDiffEqAdamsBashforthMoulton v1.9.0
[6ad6398a] OrdinaryDiffEqBDF v1.22.0
[bbf590c4] OrdinaryDiffEqCore v3.22.0
[50262376] OrdinaryDiffEqDefault v1.13.0
[4302a76b] OrdinaryDiffEqDifferentiation v2.2.1
[9286f039] OrdinaryDiffEqExplicitRK v1.9.0
[e0540318] OrdinaryDiffEqExponentialRK v1.13.0
[becaefa8] OrdinaryDiffEqExtrapolation v1.16.0
[5960d6e9] OrdinaryDiffEqFIRK v1.23.0
[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 v1.23.0
[c9986a66] OrdinaryDiffEqNordsieck v1.9.0
[5dd0a6cf] OrdinaryDiffEqPDIRK v1.11.0
[5b33eab2] OrdinaryDiffEqPRK v1.8.0
[04162be5] OrdinaryDiffEqQPRK v1.8.0
[af6ede74] OrdinaryDiffEqRKN v1.10.0
[43230ef6] OrdinaryDiffEqRosenbrock v1.25.0
[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
[90014a1f] PDMats v0.11.37
[69de0a69] Parsers v2.8.3
[ccf2f8ad] PlotThemes v3.3.0
[995b91a9] PlotUtils v1.4.4
[91a5bcdd] Plots v1.41.6
[f517fe37] Polyester v0.7.19
[1d0040c9] PolyesterWeave v0.2.2
[2dfb63ee] PooledArrays v1.4.3
[d236fae5] PreallocationTools v1.1.2
[aea7be01] PrecompileTools v1.3.3
[21216c6a] Preferences v1.5.2
[08abe8d2] PrettyTables v3.2.3
[27ebfcd6] Primes v0.5.7
[43287f4e] PtrArrays v1.4.0
[1fd47b50] QuadGK v2.11.2
[8a4e6c94] QuasiMonteCarlo v0.3.5
[3cdcf5f2] RecipesBase v1.3.4
[01d81517] RecipesPipeline v0.6.12
[731186ca] RecursiveArrayTools v3.48.0
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[79098fc4] Rmath v0.9.0
[7e49a35a] RuntimeGeneratedFunctions v0.5.17
[94e857df] SIMDTypes v0.1.0
[0bca4576] SciMLBase v2.150.0
[19f34311] SciMLJacobianOperators v0.1.12
[a6db7da4] SciMLLogging v1.9.1
[c0aeaf25] SciMLOperators v1.15.1
[431bcebd] SciMLPublic v1.0.1
[53ae85a6] SciMLStructures v1.10.0
[7e506255] ScopedValues v1.6.0
[6c6a2e73] Scratch v1.3.0
[91c51154] SentinelArrays v1.4.9
[efcf1570] Setfield v1.1.2
[992d4aef] Showoff v1.0.3
[777ac1f9] SimpleBufferStream v1.2.0
[727e6d20] SimpleNonlinearSolve v2.11.0
[ed01d8cd] Sobol v1.5.0
[a2af1166] SortingAlgorithms v1.2.2
[9f842d2f] SparseConnectivityTracer v1.2.1
[0a514795] SparseMatrixColorings v0.4.25
[276daf66] SpecialFunctions v2.7.1
[860ef19b] StableRNGs v1.0.4
[aedffcd0] Static v1.3.1
[0d7ed370] StaticArrayInterface v1.9.0
[90137ffa] StaticArrays v1.9.18
[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
[892a3eda] StringManipulation v0.4.4
[ec057cc2] StructUtils v2.7.1
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[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.12.1
[62fd8b95] TensorCore v0.1.1
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[3bb67fe8] TranscodingStreams v0.11.3
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[5c2747f8] URIs v1.6.1
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[41fe7b60] Unzip v0.2.0
[ea10d353] WeakRefStrings v1.4.2
[76eceee3] WorkerUtilities v1.6.1
[fdbf4ff8] XLSX v0.10.4
[49080126] ZipArchives v2.6.0
[a5390f91] ZipFile v0.10.1
[6e34b625] Bzip2_jll v1.0.9+0
[83423d85] Cairo_jll v1.18.5+1
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[d2c73de3] GR_jll v0.73.24+0
[b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
[7746bdde] Glib_jll v2.86.3+0
[3b182d85] Graphite2_jll v1.3.15+0
[2e76f6c2] HarfBuzz_jll v8.5.1+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.1.4+0
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.0.1+0
[1d63c593] LLVMOpenMP_jll v18.1.8+0
[dd4b983a] LZO_jll v2.10.3+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 v2.41.3+0
[89763e89] Libtiff_jll v4.7.2+0
[38a345b3] Libuuid_jll v2.41.3+0
[856f044c] MKL_jll v2025.2.0+0
[c8ffd9c3] MbedTLS_jll v2.28.1010+0
[e7412a2a] Ogg_jll v1.3.6+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.57.0+0
⌅ [30392449] Pixman_jll v0.44.2+0
[c0090381] Qt6Base_jll v6.10.2+1
[629bc702] Qt6Declarative_jll v6.10.2+1
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.1+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`
Julia Version 1.12.5
Commit 5fe89b8ddc (2026-02-09 16:05 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
OS: Windows (x86_64-w64-mingw32)
CPU: 16 × AMD Ryzen 7 5700U with Radeon Graphics
WORD_SIZE: 64
LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
GC: Built with stock GC
Threads: 16 default, 1 interactive, 16 GC (on 16 virtual cores)
Environment:
JULIA_EDITOR = code
JULIA_VSCODE_REPL = 1
Additional context
Add any other context about the problem here.
For solving a two-point boundary value problem using SciML FIRK solvers (RadauIIa5, in this case) I find that the solution process does not treat the initial guess values properly, either by using a vector initial guess or a previously calculated solution object. As shown in the accompanying MWE output, the boundary condition function at t=0 (function bca! in the MWE shown here) is first called, and it uses the proper values of the state variables from the initial guess. Then the function bcb!, at t=1, is called for the first time and it uses the state variables’ guessed values at t=0, that is, the values for the wrong boundary condition. It takes the 6th call to bcb! for the proper values for t=1 to appear.
This issue makes solution of a complex, nonlinear two-point BVP by the continuation method impractical, since no matter how close the guess is to the converged solution, the use of incorrect starting values in the t=1 boundary condition function may give potentially exception-producing results.
Minimal Reproducible Example 👇
Description of the MWE: A simple, linear, thermal problem of a bar of material extending from t=0 to t=1, with uniform internal heat generation. At t=0 the temperature is fixed at 100 degrees C and the axial heat flow is zero. At t=1 the heat flow is convective with an unknown coefficient of convection and an environmental temperature of 20 degrees C. I use a FIRK solver (the same method as in my actual, highly nonlinear problem) to solve for the temperature distribution and the convective coefficient.
In the first call to the RaduaIIa5 solver I use a vector of vectors as the initial guess. These values are from a converged case. In the second call to the solver I use a solution (sol1) object as the initial guess. I print the value part of the dual numbers that represent the three state variables (temperature, heat flux, and convective coefficient) for every time the boundary condition functions are called.
Error & Output⚠️
Note the output from the first line beginning with "t = 1.0"
Environment (please complete the following information):
using Pkg; Pkg.status()using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)versioninfo()Additional context
Add any other context about the problem here.