Fix MIRKN nlprob u0 inference + add issue 484 regression test

PR #473 already landed the equivalent fixes for MIRK and FIRK
(`copy(vec(cache.y₀))` in `__perform_*_iteration`, plus mesh-selection
and resize fixes), but missed the same `vec(::VectorOfArray)`
inference cliff in MIRKN's `__perform_mirkn_iteration`.

Under RecursiveArrayTools v4, `vec(::VectorOfArray)` returns a
`Base.ReshapedArray{T, 1, VectorOfArray{...}, …}` instead of a plain
`Vector`. NonlinearSolve's polyalg cannot infer `T, N, uType, R` of
the resulting `NonlinearSolution` when `u0` has that shape, so
`@inferred solve(::SecondOrderBVProblem, ::MIRKN, …)` would widen
the result type the same way it did for MIRK before #473. Mirror
PR #473's fix:

    nlprob = __construct_nlproblem(cache, vec(cache.y₀), copy(cache.y₀))
    →
    nlprob = __construct_nlproblem(cache, copy(vec(cache.y₀)), copy(cache.y₀))

Bumps `BoundaryValueDiffEqMIRKN` patch to 1.15.1.

Also adds a regression `@testitem` for issue #484 (the original
adaptive-mesh-refinement `UndefRefError`) using the torus-geodesic
BVP from the issue.

Co-Authored-By: Chris Rackauckas <accounts@chrisrackauckas.com>

Author: ChrisRackauckas

lib/BoundaryValueDiffEqMIRK/test/mirk_basic_tests.jl

@@ -712,3 +712,39 @@ end
     sol2 = solve(bvp2, MIRK4(), dt = 0.1, adaptive = false, nlsolve_kwargs = (; maxiters = 0))
     @test sol2.u == u_guess
 end
+
+# https://github.com/SciML/BoundaryValueDiffEq.jl/issues/484
+# Adaptive mesh refinement on a sufficiently nonlinear BVP must not throw
+# UndefRefError. The torus geodesic system is smooth and nonlinear enough
+# that the solver decides to refine the mesh.
+@testitem "Adaptive mesh refinement (issue 484)" begin
+    using BoundaryValueDiffEqMIRK
+
+    R = 3.0
+    r = 2.0
+
+    function f!(du, u, p, t)
+        sinθ, cosθ = sincos(u[1])
+        R_θ = R + r * cosθ
+        du[1] = u[3]
+        du[2] = u[4]
+        du[3] = -u[4]^2 * R_θ * sinθ / r
+        du[4] = 2 * r * sinθ / R_θ * u[3] * u[4]
+    end
+
+    a1 = [0.5, -1.2]
+    a2 = [-0.5, 0.3]
+
+    function bc!(residual, u, p, t)
+        ua = u(0.0)
+        ub = u(1.0)
+        residual[1:2] = ua[1:2] - a1
+        residual[3:4] = ub[1:2] - a2
+    end
+
+    prob = BVProblem(f!, bc!, vcat(a1, zero(a1)), (0.0, 1.0))
+    sol = solve(prob, MIRK4(); dt = 0.05)
+    @test SciMLBase.successful_retcode(sol)
+    @test sol.u[1][1:2] ≈ a1 atol = 1.0e-8
+    @test sol.u[end][1:2] ≈ a2 atol = 1.0e-8
+end

lib/BoundaryValueDiffEqMIRKN/Project.toml

@@ -1,7 +1,7 @@
 name = "BoundaryValueDiffEqMIRKN"
 uuid = "9255f1d6-53bf-473e-b6bd-23f1ff009da4"
 authors = ["Qingyu Qu <erikqqy123@gmail.com>"]
-version = "1.15.0"
+version = "1.15.1"
 
 [deps]
 ADTypes = "47edcb42-4c32-4615-8424-f2b9edc5f35b"

lib/BoundaryValueDiffEqMIRKN/src/mirkn.jl

@@ -137,7 +137,7 @@ function SciMLBase.solve!(cache::MIRKNCache{iip, T}) where {iip, T}
 end
 
 function __perform_mirkn_iteration(cache::MIRKNCache)
-    nlprob = __construct_nlproblem(cache, vec(cache.y₀), copy(cache.y₀))
+    nlprob = __construct_nlproblem(cache, copy(vec(cache.y₀)), copy(cache.y₀))
     solve_alg = __concrete_solve_algorithm(nlprob, cache.alg.nlsolve, cache.alg.optimize)
     kwargs = __concrete_kwargs(
         cache.alg.nlsolve, cache.alg.optimize, cache.nlsolve_kwargs, cache.optimize_kwargs,