Finite difference method

Abstract: Existing methods for the analysis of transient flows in pipe networks are often geared towards certain types of flows such as gas flows vis-a-vis liquid flows or isothermal flows vis-a-vis non-isothermal flows. Also, simplifying assumptions are often made which introduce inaccuracies when the method is applied outside the domain for which it was originally intended. This paper describes an implicit finite difference method based on the simultaneous pressure correction approach which is valid for both liquid and gas flows, for both isothermal and non-isothermal flows and for both fast and slow transients. The problematic convective acceleration term in the momentum equation, often neglected in other methods, is retained but eliminated by casting the momentum equation in an alternative form. The accuracy and stability of the method, depending on a time-step weighing factor α, are illustrated by analysing fast transients in a pipeline and simple branching network. The proposed method compares very well with the second-order-method of characteristics and the two-step Lax–Wendroff method. The advantages of the present method is its speed over a range of problems including both fast and slow transients, its accuracy, its stability and its flexibility. Copyright © 2001 John Wiley & Sons, Ltd.

Abstract: We consider a finite difference approximation to an inverse problem of determining an unknown source parameter p(t) which is a coefficient of the solution u in a linear parabolic equation subject to the specification of the solution u at an internal point along with the usual initial boundary conditions. The backward Euler scheme is studied and its convergence is proved via an application of the discrete maximum principle for a transformed problem. Error estimates For u and p involve numerical differentiation of the approximation to the transformed problem. Some experimental numerical results using the newly proposed numerical procedure are discussed.