A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems
Artikel i vetenskaplig tidskrift, 2011

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.


time-stepping methods

time-dependent boundary conditions



fluid problems


Method of Lines

runge-kutta methods


P. E. J. Vos

Flemish Institute for Technological Research

Imperial College London

Claes Eskilsson


A. Bolis

Imperial College London

S. Chun

Imperial College London

R. M. Kirby

University of Utah

S. J. Sherwin

Imperial College London

International Journal of Computational Fluid Dynamics

1061-8562 (ISSN) 1029-0257 (eISSN)

Vol. 25 107-125