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.
|Number of pages||19|
|Journal||International Journal of Computational Fluid Dynamics|
|Publication status||Published - 2011 Mar|
Bibliographical noteFunding 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