An overview of research on Eulerian-Lagrangian localized adjoint methods (ELLAM)

For problems of convection-diffusion type, Eulerian-Lagrangian localized adjoint methods provide a methodology that maintains the accuracy and efficiency of Eulerian-Lagrangian methods, while also conserving mass and systematically treating any type of boundary condition. In groundwater hydrology, this framework is useful for solute transport, as well as vadose-zone transport, multiphase transport,and reactive flows. The formulation was originated around 1990 by the authors, Herrera and Ewing, in a paper that appeared in Advances in Water Resources [Adv. Water Resour. 13 (1990) 187]. This paper reviews the progress in the development, analysis, and application of these methods since 1990, and suggests topics for future work.