Fix deprecation warnings

[ChrisRackauckas-Claude]

Summary

• Properly deprecate dualcache using Base.depwarn instead of a silent const alias. Now dualcache(...) emits a deprecation warning directing users to use DiffCache instead.
• Eliminate method overwrite warnings in tests by giving unique names to redefined functions (foo → foo_lbc/foo_larray in core_odes.jl, objfun → objfun2 in core_nesteddual.jl).

Details

dualcache deprecation

The previous code had:

# Legacy deprecate later
const dualcache = DiffCache

This allowed dualcache(...) to work silently with no deprecation notice, even with --depwarn=error. The fix replaces this with a proper deprecation function:

function dualcache(args...; kwargs...)
    Base.depwarn("`dualcache` is deprecated, use `DiffCache` instead.", :dualcache)
    return DiffCache(args...; kwargs...)
end

Method overwrite warnings

Tests defined the same function name (foo, objfun) multiple times in the same @safetestset module, causing WARNING: Method definition ... overwritten messages. While not deprecation errors, these warnings are noise that can mask real issues. Fixed by using unique function names.

Upstream deprecation status

Ran full test suite with --depwarn=error on Julia 1.12.4. No deprecation warnings from upstream dependencies (ArrayInterface, Adapt, SciMLBase, ForwardDiff, OrdinaryDiffEq, Optimization, etc.) were triggered. All current API usages are non-deprecated.

Test plan

• All 12 test suites pass with --depwarn=error
• No method overwrite warnings
• dualcache properly emits deprecation warning with --depwarn=yes
• dualcache properly throws error with --depwarn=error
• Aqua.jl quality checks pass (including test_undefined_exports)
• Runic formatting verified

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