An Alternative Direction Iterative Method With Second-Order Upwind Scheme For Convection-Diffusion Equations

Abstract

Consider the following convection diffusion equation, $$\\left\\{\\matrix{{\\displaystyle{\\partial u \\over \\partial t}} + b_{1}(x, y){\\displaystyle{\\partial u \\over \\partial x}} + b_{2}(y){\\displaystyle{\\partial u \\over \\partial y}} - \\left(a_{1} {\\displaystyle{\\partial^{2}u \\over \\partial x^{2}}} + a_{2} {\\displaystyle{\\partial^{2}u \\over \\partial y^{2}}}\\right) = f\\hfill \\hbox{in } \\Omega \ imes J,\\hfill\\cr u(x, y, t) = \\varphi (x, t)\\hfill \\hbox{on } \\partial \\Omega \ imes J,\\hfill\\cr u(x, y, 0) = u_{0}(x, y)\\hfill \\hbox{in } \\Omega,\\hfill\\cr}\\right. \\eqno(1)$$ where z =(0, 1) 2 (0, 1), J =(0, T ), b 1 ( x , y ), b 2 ( y ) are smooth functions and a 1 , a 2 are positive constants. When convection dominates diffusion, i.e. 0< a 1 , a 2 $\\ll$ | b |, the general finite difference or finite element methods often result in numerical oscillation [1]. The upwind method is an efficient method but is only first order accurate. For one dimensional stable problem with constant coefficient, [3] presented a high order upwind scheme, but it is difficult to extend the method to variable coefficient problem and two dimensional problem. In this paper we give an alternative direction iterative method combining with one dimensional second order upwind scheme for two dimensional problem. It can be as a high speed algorithm on parallel computer. The maximum principle and the second order uniform norm error estimate are obtained. Finally we give some numerical examples.