SUMMARYWe consider the numerical solution of a model one-dimensional diff usion-convection equation by a variety of explicit finite difference methods including conventional central and upwind replacements of the convection terms. We discuss commonly observed phenomena such as instability, unwanted oscillations in the numerical solution, and numerical diffusion and we present an analysis of these effects by simple mathematical techniques.

SUMMARYSome recently developed implicit time discretizations are discussed whose main application is to solving ordinary differential equations arising from finite element approximations to partial differential equations. Their theoretical properties, computer implementation and numerical behaviour, as observed in tests on simple examples, are compared with well-known discretizations such as the Crank-Nicholson method. INTRODUCI'IONIn the numerical solution of partial differential equations many techniques have been suggested for integrating forwards in time, ranging from simple finite difference methods such as the Crank-Nicolson method to sophisticated finite element methods in time. In the present paper time discretizations introduced by Nbrsett' are evaluated with emphasis on simplicity of implementation in computer programs. The new methods are shown to be competitive with Crank-Nicolson.Our sole concern is time discretization. Assuming that a partial differential equation has been discretized in its space dimensions to give a system of ordinary differential equations, it is this system which we consider. The fact that such a separation of variables is performed rules out of consideration some well known difference schemes as the Dufort-Frankel method which is not derived in this way.To be specific, consider the diffusion convection equation defined for 0 < x < 1 and positive time t with initial and boundary conditions:

Abstract.We consider the properties of a class of rational approximations to e -z due to Norsett. In particular we investigate the acceptability of these approximations.