Efficiency of the Eulerian–Lagrangian localized adjoint method for solving advection–dispersion equations on highly heterogeneous media

Abstract

SUMMARYThe Eulerian–Lagrangian Localized Adjoint Method (ELLAM) is well adapted for advection‐dominated transport problems. The method can use large time steps because the advection term is accurately approximated without any CFL restriction.A new formulation of ELLAM was developed by Younes et al. (Adv. Water Resour. 2006; 29:1056–1074) for unstructured triangular meshes. This formulation uses only strategic numerical integration points and thus requires a very limited number of integration points (usually 1 per element). To avoid numerical dispersion due to interpolation when several time steps are used, the method continuously tracks characteristics, and only changes due to dispersion are interpolated at each time step.In this work, we study the applicability and efficiency of this formulation for highly heterogeneous domains. The results show that (i) this formulation remains efficient even for highly heterogeneous domains, (ii) special care must be taken for the approximation of the dispersive term when large time steps are used, (iii) interpolation at intermediate times can be used to reduce memory requirement for problems with a large number of elements and small time steps, and (iv) the method can be used successfully for heterogeneous problems including injection and pumping wells, and remains much more efficient than the discontinuous Galerkin finite element method even when small time steps are required. Copyright © 2011 John Wiley & Sons, Ltd.