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Unconditionally Stable Explicit Finite Difference (USED) Schemes for Four Problems in Hydraulics

Lookup NU author(s): Dr Caspar HewettORCiD, Dr Vedrana Kutija


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Since the advent of fast digital computers there has been ever increasing use of numerical methods to solve problems in physics and engineering which are not open to analytical solution. In particular, the finite difference method has proved popular for the solution of many initial value problems that arise in hydraulic engineering, and is preferred by many practitioners because of the strong resemblance between the difference equations which are the basis of the method and the partial differential equations which describe the system being modelled. Finite difference schemes can be subdivided into two classes: explicit and implicit. Explicit schemes are characterised by calculating values of the dependent variables at one time level entirely in terms of values calculated previously, making them relatively easy to implement. However, they tend to require restrictions on the length of the time step to avoid numerical instability, and this can be expensive in terms of computation time. Implicit schemes are characterised by calculating values of the dependent variables simultaneously so that a system of equations must be solved at each time level. This leads to complicated code and can also be computationally expensive. However, implicit schemes tend to have much better stability properties than their explicit counterparts and so have been preferred for many commercial applications. This paper describes a class of finite difference scheme that has the advantages of both explicit and implicit schemes. Its explicit formulation makes implementation straightforward, while the method of calculating spatial derivatives breaks the connection between the time and space interval lengths, leading to unconditionally stable schemes. The application of these unconditionally stable explicit finite difference (USED) schemes to four problems in hydraulics is presented: heat conduction/diffusion, advection, transport and diffusion and free surface flow.

Publication metadata

Author(s): Hewett CJM, Kutija V

Editor(s): Falconer RA; Lin B; Harris EL; Wilson CAME; Cluckie ID; Han D; Davis JP; Heslop S

Publication type: Conference Proceedings (inc. Abstract)

Publication status: Published

Conference Name: Fifth International Conference on Hydroinformatics 2002

Year of Conference: 2002

ISSN: 9781843390213

Publisher: IWA Publishing