Handle FIRK expanded AD matvec fallbacks

[ChrisRackauckas-Claude]

This PR should be ignored until reviewed by @ChrisRackauckas.

Stacked on #552. Until #552 is merged, GitHub's PR diff will include those prerequisite commits; the new change here is the top commit c72455db.

Summary

• Use FIRK-local scalar interpolation matvecs for polynomial interpolation coefficient products used by interpolation boundary conditions.
• On Julia 1.12+, route expanded FIRK Enzyme/Mooncake collocation and whole-residual AD modes through AutoForwardDiff, while leaving standard-problem BC AD modes intact.
• Keep the fallback local to FIRK paths so ordinary Core/BLAS matvecs used by nested FIRK convergence tests are not redirected.
• Derive FIRK interpolation matvec output types from multiplication instead of promote_type, avoiding BigFloat/SparseConnectivityTracer.Dual promotion ambiguity in Misc tests.

Fixes #554.

Local verification

Targeted NESTED LobattoIIIc4 convergence repro on Julia 1.12.6 after keeping Core matvecs on BLAS:

(i = 9, order = 5.851916060671115, errors = [5.10702591327572e-14, 2.8337332480532496e-12, 1.7036194677189087e-10])
(i = 10, order = 5.851916060671115, errors = [5.10702591327572e-14, 2.8319568912138493e-12, 1.7036194677189087e-10])
real 330.36

Full FIRK AD test group on Julia 1.12.6:

FIRK Expanded AD Tests | 9 pass, total 9, 9m42.8s
Testing BoundaryValueDiffEqFIRK tests passed
real 1135.61

Full root Misc test group on Julia 1.10.11 after the product-type fix:

BigFloat Tests | 7 pass, total 7, 3m59.1s
Testing BoundaryValueDiffEq tests passed
real 2904.12

Fresh-resolved targeted multipoint Enzyme repros on Julia 1.13.0-rc1 after the product-type fix:

array retcode=Success
interp retcode=Success
real 432.25

Formatting/checks:

Runic --check lib/BoundaryValueDiffEqFIRK/src/adaptivity.jl
Runic --check lib/BoundaryValueDiffEqCore/src/utils.jl
passed

git diff --check
passed

[github-actions]

Benchmark Results (Julia v1.11)

Time benchmarks

	master	c72455d...	master / c72455d...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.66 ± 0.013 s	0.663 ± 0.007 s	0.996 ± 0.022
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	13.4 ± 0.22 ms	13.4 ± 0.28 ms	0.998 ± 0.026
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	2.84 ± 0.1 ms	2.89 ± 0.1 ms	0.982 ± 0.049
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	3.46 ± 0.13 ms	3.48 ± 0.11 ms	0.994 ± 0.049
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	1.65 ± 0.12 ms	1.67 ± 0.097 ms	0.984 ± 0.094
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	1.65 ± 0.16 ms	1.64 ± 0.11 ms	1.01 ± 0.12
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	3.32 ± 0.3 ms	3.4 ± 0.24 ms	0.979 ± 0.11
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0437 ± 0.0048 s	0.0438 ± 0.005 s	0.998 ± 0.16
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.0694 ± 0.0022 s	0.0702 ± 0.002 s	0.988 ± 0.042
Simple Pendulum/IIP/Shooting(Tsit5())	0.187 ± 0.085 ms	0.187 ± 0.088 ms	1 ± 0.65
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.846 ± 0.012 s	0.874 ± 0.0076 s	0.968 ± 0.016
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	18 ± 8.2 ms	18.1 ± 8.2 ms	0.997 ± 0.64
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	3.56 ± 0.23 ms	3.62 ± 0.25 ms	0.984 ± 0.093
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	4.37 ± 0.29 ms	4.42 ± 0.27 ms	0.988 ± 0.09
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	2.03 ± 0.2 ms	2.08 ± 0.2 ms	0.974 ± 0.13
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	3.98 ± 1.1 ms	3.98 ± 3.2 ms	1 ± 0.85
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	7.71 ± 6.5 ms	7.57 ± 6.2 ms	1.02 ± 1.2
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.111 ± 0.0056 s	0.111 ± 0.0047 s	0.998 ± 0.066
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.172 ± 0.0087 s	0.175 ± 0.0086 s	0.984 ± 0.069
Simple Pendulum/OOP/Shooting(Tsit5())	0.651 ± 0.12 ms	0.655 ± 0.14 ms	0.994 ± 0.28
time_to_load	8.06 ± 0.029 s	8.2 ± 0.017 s	0.983 ± 0.0041

Memory benchmarks

	master	c72455d...	master / c72455d...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.389 M allocs: 0.0443 GB	0.389 M allocs: 0.0443 GB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	0.0436 M allocs: 4.88 MB	0.0436 M allocs: 4.88 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	15.7 k allocs: 1.65 MB	15.7 k allocs: 1.65 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	22.1 k allocs: 2.03 MB	22.1 k allocs: 2.03 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	12.8 k allocs: 1.05 MB	12.8 k allocs: 1.05 MB	1
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	25.5 k allocs: 1.82 MB	25.5 k allocs: 1.82 MB	1
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	0.049 M allocs: 3.39 MB	0.049 M allocs: 3.39 MB	1
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.553 M allocs: 0.0535 GB	0.553 M allocs: 0.0535 GB	1
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.833 M allocs: 0.0778 GB	0.833 M allocs: 0.0778 GB	1
Simple Pendulum/IIP/Shooting(Tsit5())	4.63 k allocs: 0.224 MB	4.63 k allocs: 0.224 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.89 M allocs: 0.984 GB	0.89 M allocs: 0.984 GB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	0.0932 M allocs: 24.8 MB	0.0932 M allocs: 24.8 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	0.0324 M allocs: 3.96 MB	0.0324 M allocs: 3.96 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	0.045 M allocs: 4.99 MB	0.045 M allocs: 4.99 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	25.3 k allocs: 2.17 MB	25.3 k allocs: 2.17 MB	1
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	0.142 M allocs: 10.2 MB	0.142 M allocs: 10.2 MB	1
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	0.266 M allocs: 18.7 MB	0.266 M allocs: 18.7 MB	1
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	2.52 M allocs: 0.279 GB	2.52 M allocs: 0.279 GB	1
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	3.82 M allocs: 0.404 GB	3.82 M allocs: 0.404 GB	1
Simple Pendulum/OOP/Shooting(Tsit5())	0.0373 M allocs: 1.69 MB	0.0373 M allocs: 1.69 MB	1
time_to_load	0.159 k allocs: 11.2 kB	0.159 k allocs: 11.2 kB	1