An analytic adjoint Trefftz method for solving the singular parabolic convection–diffusion equation and initial pollution profile problem

Abstract

In this paper we develop a novel method for the solution of a singularly perturbed convection–diffusion equation by inserting the adjoint Trefftz functions as test functions into the derived domain/boundary integral equation. The trial solution is expressed in terms of the Trefftz series (i.e., the spatial part of the Trefftz test functions), endowing with time-dependent expansion coefficients. With the aid of the weighting orthogonality of the Trefftz series we can find these coefficients in closed-form, which are integrals of initial condition, boundary conditions and source function. Hence, we have an analytical method to find the singular solution, very close to the exact one if more series terms are used. On the other hand, when the recovery of initial pollution profile by using the measured data at a final time is concerned, the number of the Trefftz series must be suitably chosen to overcome the ill-posed behavior of the initial pollution profile problem. The present results are better than that obtained by the Lie-group shooting method.