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Science Forum Index » Math - Numerical Analysis Forum » solving extremely stiff system of ode...
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 Posted: Wed May 14, 2008 5:38 am |
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Guest
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Dear all,
Suppose we want to solve a system of odes comprising a problem with
extremely different time scales, which makes the ode system extremely
stiff.
What are possible solution strategies to this problem to obtain the
time response? E.g. a combined problem of order of Hz and of order of
GHz? I guess normal solvers, like predictor corrector need an
extremely small timestep to solve this.
Are there possibly methods to transform the stiff M C K matrices to
obtain a nonstiff problem, or is there really no free lunch? I'm
looking for the most preforming solving procedure to tackle this kind
of problem.
When solving problems in frequency domain, like FEM methods, for
higher frequency - problems we have to switch to BEM and even wave
based methods in very high frequency range. But are there suitable
strategies for time domain integration?
Any suggestion is greatly acknowledged.
kind regards,
kizzie |
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| Peter Spellucci... |
| Posted: Fri May 16, 2008 4:45 am |
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In article <a245460a-fa09-47d9-bd61-1e8280841d0a at (no spam) b64g2000hsa.googlegroups.com>,
kizzienova at (no spam) gmail.com writes:
Quote: Dear all,
Suppose we want to solve a system of odes comprising a problem with
extremely different time scales, which makes the ode system extremely
stiff.
What are possible solution strategies to this problem to obtain the
time response? E.g. a combined problem of order of Hz and of order of
GHz? I guess normal solvers, like predictor corrector need an
extremely small timestep to solve this.
methods which are A- and L-stable like Euler backwards, BDF2 and Radau5 don't
suffer from this, but there is not only the problem of stability, you also might
have a problem of attainable precision: if artificial damping is too strong
then the slow varying component might be insuvfficiently accurate for
larger time steps
if you can clearly identify the slow and fast components, then there
is a chance to get a scheme adapted to this separation:
there is a vast literature on this subject, for example
Weiner, R.; Bruder, J.
Partitioned adaptive Runge-Kutta methods for the solution of stiff and
nonstiff differential equations. (Russian)
[A] Numerical and applied mathematics, Interuniv. Sci. Collect., Ufa, 75-84
(1988).
The application of adaptive Runge-Kutta methods is considered for two
important special cases of decomposition of the original system:
1) the stiff components are known and are contained in a subsystem of
dimension $N\ll n$ (n dimension of the system),
2) a method for the whole system is chosen automatically depending on its
type (a stiff method for a stiff system, a non-stiff method for a non-stiff
system).
Dieci, Luca; Estep, Donald
Some stability aspects of schemes for the adaptive integration of
stiff initial value problems. (English)
SIAM J. Sci. Stat. Comput. 12, No.6, 1284-1303 (1991).
MSC2000: *65L20 65L06 34A34, Reviewer: J.D.P.Donnelly (Oxford)
Jannelli, Alessandra; Fazio, Riccardo
Adaptive stiff solvers at low accuracy and complexity. (English)
[J] J. Comput. Appl. Math. 191, No. 2, 246-258 (2006). ISSN 0377-0427
Summary: This paper is concerned with adaptive stiff solvers at low
accuracy and complexity for systems of ordinary differential equations.
The considered stiff solvers are: two second order Rosenbrock methods
with low complexity, and the backward differentiation formula (BDF)
method of the same order. For the adaptive algorithm we propose to use a
monitor function defined by comparing a measure of the local variability
of the solution times the used step size and the order of magnitude of the
solution instead of the classical approach based on some local error
estimation. This simple step-size selection procedure is implemented in
order to control the behavior of the numerical solution.
It is easily used to automatically adjust the step size, as the
calculation progresses, until user-specified tolerance bounds for the
introduced monitor function are fulfilled. This leads to important
advantages in accuracy, efficiency and general ease-of-use.
At the end of the paper we present two numerical tests which show the
performance of the implementation of the stiff solvers, with the
proposed adaptive procedure.
Guenther,M.; Kvaerno, A.; Rentrop, P.:
Multirate partitioned Runge-Kutta methods. (English)
[J] BIT 41, No.3, 504-514 (2001). ISSN 0006-3835; ISSN 1572-9125
Authors' abstract: The coupling of subsystems in a hierarchical
modelling approach leads to different time constants in the dynamical
simulation of technical systems. Multirate schemes exploit
the different time scales by using different time steps for the
subsystems. The stiffness of the system or at least of some subsystems in
chemical reaction kinetics or network analysis, for example,
forbids the use of explicit integration schemes.
To cope with stiff problems, we introduce multirate schemes based on
partitioned Runge-Kutta methods which avoid the coupling between active and
latent components based on interpolating and extrapolating state variables.
Order conditions and test results for such a lower order multirate
partitioned Runge-Kutta method are presented.
[Hermann Brunner (St.John's)]
Quote:
Are there possibly methods to transform the stiff M C K matrices to
obtain a nonstiff problem, or is there really no free lunch? I'm
looking for the most preforming solving procedure to tackle this kind
of problem.
not quite realistic: if you could solve the complete eigenvalue problem
(in a costant coefficient case) at least approximately
then you could transform the problem to a nonstiff one
also model reduction methods might help (based on the same principle)
Quote:
When solving problems in frequency domain, like FEM methods, for
higher frequency - problems we have to switch to BEM and even wave
based methods in very high frequency range. But are there suitable
strategies for time domain integration?
Any suggestion is greatly acknowledged.
kind regards,
kizzie
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| Posted: Fri May 16, 2008 5:23 am |
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Dear Peter,
Thank you for your valuable references.
I'll try to find out how these techniques work. I think the problem is
separable in fast and slow components, but since I'm only a mechanical
engineer, I'm not familiar with these kind of specialized solving
techniques. Do you really think a problem with frequencies of let's
say 1 Hz and 10 Ghz can be solved with a decent accuracy and in a
realistic simulation time?
Kind regards,
kizzie |
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| Peter Spellucci... |
| Posted: Fri May 16, 2008 6:39 am |
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Guest
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In article <973e11f8-23a3-4568-8e29-01fbd893a358 at (no spam) c58g2000hsc.googlegroups.com>,
kizzienova at (no spam) gmail.com writes:
Quote: Dear Peter,
Thank you for your valuable references.
I'll try to find out how these techniques work. I think the problem is
separable in fast and slow components, but since I'm only a mechanical
engineer, I'm not familiar with these kind of specialized solving
techniques. Do you really think a problem with frequencies of let's
say 1 Hz and 10 Ghz can be solved with a decent accuracy and in a
realistic simulation time?
Kind regards,
kizzie
yes, of course by specialized techniques. In chemical engineering the situation
is sometimes even harder.
good luck
peter |
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| Posted: Mon May 19, 2008 8:59 am |
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On 16 mei, 18:39, spellu... at (no spam) fb04373.mathematik.tu-darmstadt.de (Peter
Spellucci) wrote:
Quote: In article <973e11f8-23a3-4568-8e29-01fbd893a... at (no spam) c58g2000hsc.googlegroups.com>, kizzien... at (no spam) gmail.com writes:
Dear Peter,
Thank you for your valuable references.
I'll try to find out how these techniques work. I think the problem is
separable in fast and slow components, but since I'm only a mechanical
engineer, I'm not familiar with these kind of specialized solving
techniques. Do you really think a problem with frequencies of let's
say 1 Hz and 10 Ghz can be solved with a decent accuracy and in a
realistic simulation time?
Kind regards,
kizzie
yes, of course by specialized techniques. In chemical engineering the situation
is sometimes even harder.
good luck
peter
Thanks a lot!
kizzie |
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