refactor: splitting up of NonlinearSolve #124

Job	Run time
downgrade (1.10, core)	12m 36s
test (1.10, ubuntu-latest, core)	12m 32s
downgrade (1.10, adjoint)	8m 35s
downgrade (1.10, alloc_check)	8m 19s
test (1.10, ubuntu-latest, adjoint)	8m 43s
test (1.10, ubuntu-latest, alloc_check)	8m 49s
test (1.10, macos-latest, core)	1s
test (1.10, macos-latest, adjoint)	1s
test (1.10, macos-latest, alloc_check)	1s
test (1.10, windows-latest, core)	14m 43s
test (1.10, windows-latest, adjoint)	10m 25s
test (1.10, windows-latest, alloc_check)	10m 49s
test (1, ubuntu-latest, core)	21m 30s
test (1, ubuntu-latest, adjoint)	15m 34s
test (1, ubuntu-latest, alloc_check)	15m 2s
test (1, macos-latest, core)	1s
test (1, macos-latest, adjoint)	1s
test (1, macos-latest, alloc_check)	1s
test (1, windows-latest, core)	24m 1s
test (1, windows-latest, adjoint)	17m 8s
test (1, windows-latest, alloc_check)	16m 30s
	3h 25m 22s

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