Eulerian‐Lagrangian localized adjoint method: The theoretical framework

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Abstract
This is the second of a sequence of papers devoted to applying the localized adjoint method (LAM), in space‐time, to problems of advective‐diffusive transport. We refer to the resulting methodology as the Eulerian‐Lagrangian localized adjoint method (ELLAM). The ELLAM approach yields a general formulation that subsumes many specific methods based on combined Lagrangian and Eulerian approaches, so‐called characteristic methods (CM). In the first paper of this series the emphasis was placed in the numerical implementation and a careful treatment of implementation of boundary conditions was presented for one‐dimensional problems. The final ELLAM approximation was shown to possess the conservation of mass property, unlike typical characteristic methods. The emphasis of the present paper is on the theoretical aspects of the method. The theory, based on Herrera's algebraic theory of boundary value problems, is presented for advection‐diffusion equations in both one‐dimensional and multidimensional systems. This provides a generalized ELLAM formulation. The generality of the method is also demonstrated by a treatment of systems of equations as well as a derivation of mixed methods. © 1993 John Wiley & Sons, Inc.

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