A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations

We develop a family of Eulerian-Lagrangian localized adjoint methods for th e solution of the initial-boundary value problems for first-order advection -reaction equations on general multi-dimensional domains. Different trackin g algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes, which are fully mass conservative, naturally incorporate inflow boundary conditions into their formulations and do not need any arti ficial outflow boundary conditions. Moreover, they have regularly structure d, well-conditioned, symmetric, and positive-definite coefficient matrices, which can be efficiently solved by the conjugate gradient method in an opt imal order number of iterations without any preconditioning needed. Numeric al results are presented to compare the performance of the ELLAM schemes wi th many well studied and widely used methods, including the upwind finite d ifference method, the Galerkin and the Petrov-Galerkin finite element metho ds with backward-Euler or Crank-Nicolson temporal discretization, the strea mline diffusion finite element methods, the monotonic upstream-centered sch eme for conservation laws (MUSCL). and the Minmod scheme. (C) 1999 Academic Press.