The partial differential equations used to model multiphase and multicomponent fluid flows are convection-dominated, with important local properties. Operator-splitting techniques have been defined to address these different phenomena. Convection is treated by time stepping along the characteristics of the associated pure convection problem and diffusion is modeled via a Galerkin method for miscible displacement and a Petrov Galerkin method for immiscible displacement. These ideas have been generalized to Eulerian-Lagrangian Localized Adjoint Method (ELLAM) formulations which conserve mass and allow more accurate treatment of boundary conditions. Accurate approximations of the fluid velocities needed in the characteristic time stepping are obtained by mixed finite-element methods.


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