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 the solution of the initial-boundary value problems for rst-order advection-reaction equations on general multi-dimensional domains. Diierent tracking algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes naturally incorporate innow boundary conditions into their formulations and do not need any artii-cial outtow boundary condition. They are fully mass conservative. Moreover, they have regularly structured, well-conditioned, symmetric and positive-deenite coeecient matrices, which can be solved eeciently by, for example, the conjugate gradient method in an optimal order number of iterations without any preconditioning needed. Numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the upwind nite diierence method, the Galerkin and the Petrov-Galerkin nite element methods with backward-Euler or Crank-Nicolson temporal discretization, and the streamline diiusion nite element methods.