A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems

Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework - both in terms of algorithms and implementations - that allows an almost seamless switch between v...

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Bibliographic Details
Published in:International journal of computational fluid dynamics Vol. 25; no. 3; pp. 107 - 125
Main Authors: Vos, Peter E.J., Eskilsson, Claes, Bolis, Alessandro, Chun, Sehun, Kirby, Robert M., Sherwin, Spencer J.
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 01.03.2011
Taylor & Francis Ltd
Subjects:
Algorithms
Computational fluid dynamics
Design engineering
dimsims
Fluid dynamics
Fluid flow
fluid problems
Fluids
Generalized linear models
IMEX-schemes
implementation
Mathematical models
Method of Lines
Object oriented
object-oriented implementation
Object-oriented programming
Partial differential equations
runge-kutta methods
time-dependent boundary conditions
time-stepping methods
ISSN:1061-8562, 1029-0257, 1029-0257
Online Access:Get full text
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Summary:Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework - both in terms of algorithms and implementations - that allows an almost seamless switch between various explicit, implicit and implicit-explicit (IMEX) time-stepping methods. We put particular emphasis on how to incorporate time-dependent boundary conditions, an issue that goes beyond classical ODE theory but which plays an important role in the time-stepping of the PDEs arising in computational fluid dynamics. Our algorithm is based upon J.C. Butcher's unifying concept of general linear methods that we have extended to accommodate the family of IMEX schemes that are often used in engineering practice. In the article, we discuss design considerations and present an object-oriented implementation. Finally, we illustrate the use of the framework by applications to a model problem as well as to more complex fluid problems.
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ISSN:1061-8562
1029-0257
1029-0257
DOI:10.1080/10618562.2011.575368