Adjoint methods are particle methods: Implications for Eulerian-Lagrangian
modeling of multiphase multicomponent transport

Thomas F. Russell
Division of Mathematical Sciences
National Science Foundation

The Eulerian-Lagrangian localized adjoint method (ELLAM) was
originally developed for linear advection-diffusion equations.  It has
no CFL condition, handles all kinds of boundary conditions rigorously,
is fully mass-conservative, and its Lagrangian step symmetrizes the
advection-diffusion operator.  We present here a natural adjoint
formulation that extends ELLAM to multiphase multicomponent flows with
nonlinear fluxes.  This is based on a physical interpretation of the
dual adjoint operator as a propagator of mass-carrying particles, in
contrast to the wave-propagating primal direct operator.  The adjoint
operator is linear, so that its space-time characteristics do not
intersect, shocks do not form, and particles do not disappear.  This
makes the dual framework substantially more tractable computationally.
All of these concepts are illustrated in some examples and in the
discretization of a compositional model.  This represents joint work
with B.O. Heimsund, H.K. Dahle, and M.S. Espedal of the University of
Bergen, Norway.