MIRK methods failed on swirling flow III when viscosity parameter is small

[ErikQQY]

Describe the bug 🐞

$\epsilon$ is the viscosity parameter, when it is set to 0.1, we can successfully get the solution, but when we set $\epsilon$ as 0.01 or smaller, MIRK methods throw error.

Minimal Reproducible Example 👇

using BoundaryValueDiffEq
eps = 0.01
function test!(du, u, p, t)
    eps = p
    du[1] = u[2]
    du[2] = (u[1]*u[4] - u[3]*u[2])/eps
    du[3] = u[4]
    du[4] = u[5]
    du[5] = u[6]
    du[6] = (-u[3]*u[6] - u[1]*u[2])/eps
end

function bc!(res, u, p, t)
    res[1] = u[1][1] + 1.0
    res[2] = u[1][3]
    res[3] = u[1][4]
    res[4] = u[end][1] - 1.0
    res[5] = u[end][3]
    res[6] = u[end][4]
end
tspan = (0.0, 1.0)
u0 = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
prob = BVProblem(test!, bc!, u0, tspan, eps)
sol=solve(prob, MIRK4(), dt=0.01)

Error & Stacktrace ⚠️

julia> sol = solve(prob, MIRK4(), dt=0.01)
ERROR: MethodError: no method matching (::BoundaryValueDiffEq.var"#119#125"{BoundaryValueDiffEq.MIRKCache{true, Float64, Tuple{Int64}, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, Tuple{typeof(bca!), typeof(bcb!)}, BVProblem{Vector{Float64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, TwoPointBVProblem{true}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}, TwoPointBVProblem{true}, SciMLBase.NullParameters, MIRK4{Nothing, BVPJacobianAlgorithm{AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}}}, BoundaryValueDiffEq.MIRKTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}}, BoundaryValueDiffEq.MIRKInterpTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{PreallocationTools.DiffCache{Matrix{Float64}, Vector{Float64}}}, Vector{Matrix{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, Vector{Vector{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}, Vector{Float64}, Vector{Vector{Float64}}, Vector{Vector{Float64}}, Tuple{Tuple{Int64}, Tuple{Int64}}, NamedTuple{(:defect_threshold, :MxNsub, :abstol, :dt, :adaptive), Tuple{Float64, Int64, Float64, Float64, Bool}}}, TwoPointBVProblem{true}})(::Vector{Float64}, ::Vector{Float64})

Closest candidates are:
  (::BoundaryValueDiffEq.var"#119#125")(::Any, ::Any, ::Any)
   @ BoundaryValueDiffEq ~/.julia/packages/BoundaryValueDiffEq/SrG4p/src/solve/mirk.jl:208

Stacktrace:
  [1] (::NonlinearFunction{true, SciMLBase.FullSpecialize, BoundaryValueDiffEq.var"#119#125"{BoundaryValueDiffEq.MIRKCache{true, Float64, Tuple{Int64}, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, Tuple{typeof(bca!), typeof(bcb!)}, BVProblem{Vector{Float64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, TwoPointBVProblem{true}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}, TwoPointBVProblem{true}, SciMLBase.NullParameters, MIRK4{Nothing, BVPJacobianAlgorithm{AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}}}, BoundaryValueDiffEq.MIRKTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}}, BoundaryValueDiffEq.MIRKInterpTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{PreallocationTools.DiffCache{Matrix{Float64}, Vector{Float64}}}, Vector{Matrix{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, Vector{Vector{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}, Vector{Float64}, Vector{Vector{Float64}}, Vector{Vector{Float64}}, Tuple{Tuple{Int64}, Tuple{Int64}}, NamedTuple{(:defect_threshold, :MxNsub, :abstol, :dt, :adaptive), Tuple{Float64, Int64, Float64, Float64, Bool}}}, TwoPointBVProblem{true}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, BoundaryValueDiffEq.var"#147#151"{SparseDiffTools.ForwardDiffJacobianCache{SparseDiffTools.MatrixColoringResult{Vector{Int64}, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, ArrayInterface.BandedMatrixIndex, ArrayInterface.BandedMatrixIndex}, ForwardColorJacCache{Vector{ForwardDiff.Dual{ForwardDiff.Tag{BoundaryValueDiffEq.BoundaryValueDiffEqTag, Float64}, Float64, 8}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{BoundaryValueDiffEq.BoundaryValueDiffEqTag, Float64}, Float64, 8}}, Vector{Float64}, Vector{Vector{NTuple{8, Float64}}}, Vector{Int64}, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}}, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, Vector{Float64}, Vector{Float64}}, Vector{Float64}, BoundaryValueDiffEq.var"#145#149"{BoundaryValueDiffEq.MIRKCache{true, Float64, Tuple{Int64}, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, Tuple{typeof(bca!), typeof(bcb!)}, BVProblem{Vector{Float64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, TwoPointBVProblem{true}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}, TwoPointBVProblem{true}, SciMLBase.NullParameters, MIRK4{Nothing, BVPJacobianAlgorithm{AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}}}, BoundaryValueDiffEq.MIRKTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}}, BoundaryValueDiffEq.MIRKInterpTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{PreallocationTools.DiffCache{Matrix{Float64}, Vector{Float64}}}, Vector{Matrix{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, Vector{Vector{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}, Vector{Float64}, Vector{Vector{Float64}}, Vector{Vector{Float64}}, Tuple{Tuple{Int64}, Tuple{Int64}}, NamedTuple{(:defect_threshold, :MxNsub, :abstol, :dt, :adaptive), Tuple{Float64, Int64, Float64, Float64, Bool}}}, BoundaryValueDiffEq.var"#119#125"{BoundaryValueDiffEq.MIRKCache{true, Float64, Tuple{Int64}, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, Tuple{typeof(bca!), typeof(bcb!)}, BVProblem{Vector{Float64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, BVPFunction{true, SciMLBase.FullSpecialize, true, typeof(test!), Tuple{typeof(bca!), typeof(bcb!)}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing}, TwoPointBVProblem{true}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}, TwoPointBVProblem{true}, SciMLBase.NullParameters, MIRK4{Nothing, BVPJacobianAlgorithm{AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}}}, BoundaryValueDiffEq.MIRKTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Matrix{Float64}}, BoundaryValueDiffEq.MIRKInterpTableau{Int64, Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{PreallocationTools.DiffCache{Matrix{Float64}, Vector{Float64}}}, Vector{Matrix{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, Vector{Vector{Float64}}, Vector{PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}}, PreallocationTools.DiffCache{Vector{Float64}, Vector{Float64}}, Vector{Float64}, Vector{Vector{Float64}}, Vector{Vector{Float64}}, Tuple{Tuple{Int64}, Tuple{Int64}}, NamedTuple{(:defect_threshold, :MxNsub, :abstol, :dt, :adaptive), Tuple{Float64, Int64, Float64, Float64, Bool}}}, TwoPointBVProblem{true}}}, BVPJacobianAlgorithm{AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}, AutoSparseForwardDiff{nothing, BoundaryValueDiffEq.BoundaryValueDiffEqTag}}}, Nothing, Nothing, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, BandedMatrices.BandedMatrix{Float64, Matrix{Float64}, Base.OneTo{Int64}}, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED_NO_TIME), Vector{Int64}, Nothing, Vector{Float64}})(::Vector{Float64}, ::Vararg{Vector{Float64}})

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

julia> Pkg.status()
Status `~/.julia/environments/v1.9/Project.toml`
  [ded0fc24] BVProblemLibrary v0.1.4
⌃ [6e4b80f9] BenchmarkTools v1.3.2
  [764a87c0] BoundaryValueDiffEq v5.6.0
  [54ca160b] ODEInterface v0.5.0

• Output of versioninfo()

julia> versioninfo()
Julia Version 1.9.2
Commit e4ee485e909 (2023-07-05 09:39 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 8 × Intel(R) Core(TM) i5-10210U CPU @ 1.60GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-14.0.6 (ORCJIT, skylake)
  Threads: 1 on 8 virtual cores