From Backward Approximations to Lagrange Polynomials in Discrete Advection–Reaction Operators

In this work we introduce a family of operators called discrete advection–reaction operators. These operators are important on their own right and can be used to efficiently analyze the asymptotic behavior of a finite differences discretization of variable coefficient advection–reaction–diffusion partial differential equations. They consists of linear bidimensional discrete dynamical systems defined in the space of real sequences. We calculate explicitly their asymptotic evolution by means of a matrix representation. Finally, we include the special case of matrices with different eigenvalues to show the connection between the operators evolution and interpolation theory.