Breaking the Taboos: Stability of Finite Difference for the Solution of Flow problems

  1. Lookup NU author(s)
  2. Dr Casper Hewett
  3. Dr Vedrana Kutija
  4. Dr Kutsi Erduran
Author(s)Hewett CJM, Kutija V, Erduran KS
Editor(s)Cortezón JAR; de Jesús Soriano Pérez T
Publication type Conference Proceedings (inc. Abstract)
Conference NameHydroinformatics 2000
Conference LocationCedar Rapids, Iowa, USA
Year of Conference2000
Date23-27 August 2000
Volume
Pages
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In this age of fast digital computers many problems which cannot be solved by analytical methods can be approximated by numerical methods such as the finite difference method. Some success has been achieved in using finite differences to solve time dependent problems such as those described by the transport-diffusion equation and the equations for free surface flow. Explicit schemes are relatively easy to implement but have the disadvantage that they can exhibit numerical instability. Imposing time step limitations solves the problem, but the restrictions can be severe, and thus computationally expensive. On the other hand, implicit schemes do not generally exhibit stability problems, but have the disadvantage that they typically require a system of equations to be solved at each space-time grid point, which can again be computationally intensive. Stability criteria which restrict the time step are typical and linearised stability analysis using Fourier series has been developed to evaluate such criteria. The method is described and some well-known schemes are reviewed. NewC is an implicit finite difference scheme for one-dimensional free surface problems. It is capable of modelling subcritical, supercritical and transcritical flows without the modifications to the governing equations required by other schemes. It also has the advantage that it is easy to implement for network problems. Linearised stability analysis for the NewC scheme is presented. Recent work has led to the development of explicit finite difference schemes which are also unconditionally stable. This is achieved by introducing an intermediate parameter which effectively decouples the time and space dimensions. The application of such a scheme to the transport-diffusion equation is presented.
PublisherIowa Institute of Hidraulic Research