A new efficient Eulerian–Lagrangian localized adjoint method for solving the advection–dispersion equation on unstructured meshes

A new and high efficient scheme is developed for the Eulerian–Lagrangian Localized Adjoint Method (ELLAM) to solve the Advection–Dispersion transport Equation (ADE) on unstructured triangular meshes. To obtain accurate results, the new method requires a very limited number of integration points (usually 1 per element).The scheme uses only strategic points as numerical integration points. With this scheme, locations of integration points and weights are assigned at the new time level and then backtracked to the old time level without redistributing the weights. Interpolation problems are minimized since we use continuous characteristics and only changes due to dispersion are interpolated to obtain the concentration at the foot of each characteristic.Different numerical experiments with a large range of grid Peclet numbers are presented to compare the new ELLAM to the standard one and to the Discontinuous Finite Element Method. The new ELLAM gives more accurate results and is much less CPU time consuming than all other methods especially with large time steps and highly unstructured meshes.