Approaches to solving the ADE can be divided into Eulerian, Lagrangian, and Mixed Eulerian-Lagrangian methods:

Eulerian

Eulerian methods employ a fixed grid. They are convenient and efficient, however ...

When advective transport dominates the transport problem, which is the most common condition in ground water contamination settings, then Eulerian techniques lead to: errors

Simulations typically exhibit a good mass balance, but include concentrations less than zero, greater than the source strength, and inaccurate concentrations due to dispersion introduced from the numerical calculations that the modeler cannot separate from the influence of physical dispersion.

Lagrangian

Lagrangian methods employ a deforming grid or deforming coordinates in a fixed grid. They are excellent for pure advection problems, however

When dispersion is included with the advective transport, particularly in heterogeneous media with multiple sources and sinks, then Lagrangian techniques become cumbersome,

thus the:

Mixed Eulerian-Lagrangian

Mixed Eulerian-Lagrangian methods employ a fixed grid to solve the advective term of the ADE and a deforming grid or coordinates for the dispersive term of the equation.

They prevent overshoot/undershoot and reduce numerical dispersion but are

LESS efficient than Eulerian methods and can suffer from mass balance problems.