Abstract
Abstract Symbolic operations are used together with delta functions to derive the generalized adjoint method for physical processes that contain first-order discontinuities caused by parameterized on/off switches with zero-order discontinuities in the source term. Generalized adjoint solutions are obtained analytically for simple heuristic examples and verified by direct perturbation analyses. Errors due to the conventional treatment with the “classic” adjoint method (which ignores the variation of the switch point) are quantified and found to be significant. The classic adjoint method encounters more serious problems when the parameterized process causes on/off oscillations in a numerical integration of the equation. In the limit of a vanishing computational time step, the on/off oscillations approach a marginal state that can be well treated by the generalized adjoint method. It is found that the marginal state imposes a constraint on the perturbation. Three basic issues are raised and addressed concern...
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