🚀 go-pugleaf

Dear colleagues,

I am looking for some recent references comparing
the efficiencies of Runge-Kutta and Multistep methods.

My main interest is in the field of time-dependent
pseudo-hyperbolic PDEs describing advection/convection
dominated flows (i.e. diffusion/viscosity very small).
If these PDEs are discretized in space with for example
a finite element method, then a system of ODEs results
which can be solved with either a Runge-Kutta scheme or
a Multistep scheme.

Can you give me some references where the efficiencies
of those schemes are compared thoroughly ?
(Preferably for high spatial resolutions, i.e. very large
 systems of ODEs.)

Please don't give a statement such as ``Runge-Kutta schemes
are too expensive'' without giving a good reference which
proves it.  In fact, section 3.7 of the book:

   E. Hairer, S.P. Norsett, G. Wanner,
      Solving Ordinary Differential Equations I,
      Sringer Verlag, Berlin, 1987

seems to indicate that accurate Runge-Kutta schemes
(such as DOPRI-8) may outperform variable order/stepsize
Multistep schemes for a lot of problems.  Hairer et al apply
both families of schemes to small systems of ODEs as well as
a ``large'' system of ODEs resulting from space-discretization
of a reaction-diffusion problem.

Is DOPRI-8 also more efficient for advection dominated flows ???

I also find Table 7.1 of Hairer et al interesting.  It compares
the storage required for Runge-Kutta and Multistep methods.
The surprising conclusion is that DOPRI-8 is most economical...
Any comments on that ?

Thank you  &  greetings,
-----------------------------------------------------------------------------
Fred Walsteijn                                | Internet: walsteyn@fys.ruu.nl
Institute for Marine and Atmospheric Research | FAX:      31-30-543163
Utrecht University, The Netherlands           | Phone:    31-30-533169