Generalized Adjoint for Physical Processes with Parameterized Discontinuities. Part II: Vector Formulations and Matching Conditions

Abstract

Abstract Generalized tangent linear and adjoint equations are derived for a vector equation that contains a parameterized source term with discontinuous on/off switches controlled by a threshold condition. As an extension of Part I, the key results here include a pair of interface matching conditions for coupled tangent linear and adjoint vectors across a switch point. Each matching condition can be expressed in either a forward form or a backward form that connects the vector values on the two sides of the switch point via a forward- or backward-matching matrix. The forward- and backward-matching matrices are mutually invertible. The backward/forward-matching matrix for the adjoint vector is the transpose of the forward/backward matrix for the tangent linear vector. By using the matching condition, the classic tangent linear (or adjoint) solution can be extended through a switch point, so a fundamental set of generalized tangent linear (or adjoint) solutions can be constructed, which leads to an explicit...