A positivity preserving characteristic finite element method for solving the transport and convection–diffusion–reaction equations on general surfaces

Abstract

In this paper, a positivity preserving characteristic finite element method is presented to solve the transport and convection–diffusion–reaction equations on general surfaces. The surface finite element method, which solves a variation problem by the linear finite element space on a piecewise triangulated surface, is applied to spatial discretization. For the backtracking in characteristic derivative discretization, unlike the cases on the two-dimensional plane, the foots of approximate characteristics may locate in the outer domain of the surface. To determine the values of solution at the foots of characteristics, a new strategy, which permits larger time steps, is designed instead of the discrete closest point mapping method which has a strict time step restriction. By combining with the mass lumping technique, the proposed numerical scheme is positivity preserving. The proposed method can also be extended to the problems with nonlinear convection terms. Various numerical examples are performed to demonstrate the validity and accuracy of the proposed method.