A new fourth-order difference scheme for solving an<i>N</i>-carrier system with Neumann boundary conditions

A new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy, is developed by using a Padé approximation method by solving an N-carrier system with Neumann boundary conditions.

Abstract

Abstract In this paper, a numerical method is developed to solve an N-carrier system with Neumann boundary conditions. First, we apply the compact finite difference scheme of fourth order for discretizing spatial derivatives at the interior points. Then, we develop a new combined compact finite difference scheme for the boundary, which also has fourth-order accuracy. Lastly, by using a Padé approximation method for the resulting linear system of ordinary differential equations, a new compact finite difference scheme is obtained. The present scheme has second-order accuracy in time direction and fourth-order accuracy in space direction. It is shown that the scheme is unconditionally stable. The present scheme is tested by two numerical examples, which show that the convergence rate with respect to the spatial variable from the new scheme is higher and the solution is much more accurate when compared with those obtained by using other previous methods. Keywords: N-carrier systemPadé approximationNeumann boundary conditionenergy exchangedifference scheme 2000 AMS Subject Classifications : 65M0665M12 Acknowledgements The authors thank the anonymous referees for their valuable suggestions for the improvement of this paper. The first author is supported by the Natural Science Research Foundation of Universities in Anhui (KJ2011B112). The second author is supported by the Natural Science Foundation of P.R. China (10962001), Guangxi Natural Science Foundation (2010GXNSFA013115, 2011GXNSFD018006) and Hunan Key Laboratory for Computation and Simulation in Science and Engineering.