An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation

Many numerical methods use characteristic analysis to accommodate the advective component of transport. Such characteristic methods include Eulerian-Lagrangian methods (ELM), modified method of characteristics (MMOC), and operator splitting methods. A generalization of characteristic methods can be developed using an approach that we refer to as an Eulerian-Lagrangian localized adjoint method (ELLAM). This approach is a space-time extension of the optimal test function (OTF) method. The method provides a consistent formulation by defining test functions as specific solutions of the localized homogeneous adjoint equation. All relevant boundary terms arise naturally in the ELLAM formulation, and a systematic and complete treatment of boundary condition implementation results. This turns out to have significant implications for the calculation of boundary fluxes. An analysis of global mass conservation leads to the final ELLAM approximation, which is shown to possess the conservative property. Numerical calculations demonstrate the behaviour of the method with emphasis on treatment of boundary conditions. Discussion of the method includes ideas on extensions to higher spatial dimensions, reactive transport, and variable coefficient equations.