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

Peter E.J. Vos; Claes Eskilsson; Alessandro Bolis; Sehun Chun; Robert M. Kirby; Spencer J. Sherwin

doi:10.1080/10618562.2011.575368

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

Peter E.J. Vos, Claes Eskilsson, Alessandro Bolis, Sehun Chun, Robert M. Kirby, Spencer J. Sherwin

Integrated Sciences and Engineering Division

Research output: Contribution to journal › Article › peer-review

24 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	107-125
Number of pages	19
Journal	International Journal of Computational Fluid Dynamics
Volume	25
Issue number	3
DOIs	https://doi.org/10.1080/10618562.2011.575368
Publication status	Published - 2011 Mar

Bibliographical note

Funding Information:
The authors would like to acknowledge the insightful input of Professor Ray Spiteri of the University of Saskatchewan. SJS would like to acknowledge the support under an EPSRC Advanced Research Fellowship, SC would like to acknowledge support from the CardioMath initiative of the Institute of Mathematical Science at Imperial College London, and RMK would like to acknowledge support under the Leverhulme Foundation Trust.

All Science Journal Classification (ASJC) codes

Computational Mechanics
Aerospace Engineering
Condensed Matter Physics
Energy Engineering and Power Technology
Mechanics of Materials
Mechanical Engineering

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Vos, P. E. J., Eskilsson, C., Bolis, A., Chun, S., Kirby, R. M., & Sherwin, S. J. (2011). A generic framework for time-stepping partial differential equations (PDEs): General linear methods, object-oriented implementation and application to fluid problems. International Journal of Computational Fluid Dynamics, 25(3), 107-125. https://doi.org/10.1080/10618562.2011.575368

Vos, Peter E.J. ; Eskilsson, Claes ; Bolis, Alessandro ; Chun, Sehun ; Kirby, Robert M. ; Sherwin, Spencer J. / A generic framework for time-stepping partial differential equations (PDEs) : General linear methods, object-oriented implementation and application to fluid problems. In: International Journal of Computational Fluid Dynamics. 2011 ; Vol. 25, No. 3. pp. 107-125.

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abstract = "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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note = "Funding Information: The authors would like to acknowledge the insightful input of Professor Ray Spiteri of the University of Saskatchewan. SJS would like to acknowledge the support under an EPSRC Advanced Research Fellowship, SC would like to acknowledge support from the CardioMath initiative of the Institute of Mathematical Science at Imperial College London, and RMK would like to acknowledge support under the Leverhulme Foundation Trust.",

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Vos, PEJ, Eskilsson, C, Bolis, A, Chun, S, Kirby, RM & Sherwin, SJ 2011, 'A generic framework for time-stepping partial differential equations (PDEs): General linear methods, object-oriented implementation and application to fluid problems', International Journal of Computational Fluid Dynamics, vol. 25, no. 3, pp. 107-125. https://doi.org/10.1080/10618562.2011.575368

A generic framework for time-stepping partial differential equations (PDEs) : General linear methods, object-oriented implementation and application to fluid problems. / Vos, Peter E.J.; Eskilsson, Claes; Bolis, Alessandro; Chun, Sehun; Kirby, Robert M.; Sherwin, Spencer J.

In: International Journal of Computational Fluid Dynamics, Vol. 25, No. 3, 03.2011, p. 107-125.

Research output: Contribution to journal › Article › peer-review

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