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Algorithmic differentiation (AD) has become increasingly capable andstraightforward to use. However, AD is inefficient when applied directly tosolvers, a feature of most engineering analyses. We can leverage implicitdifferentiation to define a general AD rule, making adjoints automatic.Furthermore, we can leverage the structure of differential equations toautomate unsteady adjoints in a memory efficient way. We also derive atechnique to speed up explicit differential equation solvers, which have noiterative solver to exploit. All of these techniques are demonstrated onproblems of various sizes, showing order of magnitude speed-ups with minimalcode changes. Thus, we can enable users to easily compute accurate derivativesacross complex analyses with internal solvers, or in other words, automateadjoints using a combination of AD and implicit differentiation.

Automating Steady and Unsteady Adjoints: Efficiently Utilizing Implicit and Algorithmic Differentiation