Eulerian-Lagrangian localized adjoint methods for linear advection or advection-reaction equations and their convergence analysis

In this paper, we develop Eulerian-Lagrangian localized adjoint methods (ELLAM) to solve the initial-boundary value problems for linear advection or advection-reaction equations. In contrast to many methods for advection-type problems, our ELLAM scheme naturally incorporates the inflow boundary conditions into its formulations and does not need an artificial outflow boundary condition. It does conserve mass. Moreover, optimal-order error estimates for ELLAM have been obtained. In contrast, many methods have only suboptimal-order estimates when applied to solve these problems. Furthermore, our ELLAM scheme provides a systematic approach to treat the interface problems of advection-type equations and can be naturally combined with domain decomposition and local refinement techniques to solve these problems. Numerical results in one and two dimensions are presented and discussed.