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.