Picard Integral Deferred Correction Methods
Anita Layton and Michael Minion
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Many physical systems with dynamics that involve
two or more processes with widely-differing characteristic
time scales are of interest to researchers
in the physical and biological sciences.
Well-known examples include combustion,
the transport of air pollutants, and the
movement of contaminants or microorganisms
in ground water systems. In each of these
examples, the mathematical models used to describe
the dynamics consist of systems of partial differential
equations (PDEs), which specify the advection,
diffusion, and reaction of chemical species
within a moving medium.
Hence, the construction of accurate, stable, and
efficient numerical methods for the solution of
advection-diffusion-reaction (A-D-R) equations
is a topic of significant current interest.
In this study, high-order multi-implicit
spectral deferred correction (MISDC) methods are presented for solving
A-D-R equations. MISDC methods are similar to operating-splitting methods
both types of methods give rise
to multiple implicit equations, for which different time steps may be used.
In theory, however, the temporal order of accuracy of MISDC methods can be
arbitrarily high because both the integration and splitting errors
are eliminated during the deferred correction process.
MISDC methods, which are a generalization of semi-implicit spectral deferred
correction
(SISDC) methods introduced by Minion (which are in turn modifications
of the explicit and implicit spectral deferred correction methods previously
developed by Dutt et al.),
use a low-order numerical method to compute a high-order approximation.
This is achieved by using the low-order numerical method to solve a series of
correction equations, each of which increases the order of accuracy
of the approximation by one.
The accuracy and stability of MISDC methods for A-D-R equations
and its efficiency relative to SISDC methods are investigated in this study.
The above figure shows results of an efficiency comparison between MISDC
methods and semi-implicit additive Runge-Kutta (ARK) methods
developed by Kennedy et al.,
using a simple, one-dimensional model of flamelets, with a large reaction
coefficient, as a test problem.
These results indicate that for problems with stiff reactions,
MISDC methods can be constructed to be competitive with ARK methods.
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Related Publications
- Anita T. Layton and Michael L. Minion,
"Implications of the Choice of Predictors for Semi-Implicit
Picard Integral Deferred Correction Methods,"
Comm. Appl. Math. Comp. Sci., in press.
- Anne Bourlioux, Anita T. Layton, and Michael L. Minion,
``
High-order multi-implicit spectral deferred correction methods
for problems of reactive flows,''
Journal of Computational Physics, Vol. 189, No. 2, pp. 651-675, 2003.
- Anita T. Layton and Michael L. Minion,
``
Conservative multi-implicit spectral deferred correction methods
for reacting gas dynamics,''
Journal of Computational Physics, Vol. 194, No. 2, pp. 697-715, 2004.
- Anita T. Layton and Michael L. Minion,
"
Implications of the choice of quadrature nodes for Picard
integral deferred corrections
methods for ordinary differential equations,"
BIT, in press.
- Anita T. Layton,
``
High-order operator-splitting methods for reacting gas dynamics,''
Proceedings of the
11th Annual Conference of the CFD Society of Canada
, 2003.
- Anita T. Layton.
"Conservative multi-implicit spectral deferred correction methods
with adaptive mesh refinement," Proceedings of the
12th Annual Conference of the
CFD Society of Canada, 2004.
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Last updated: September 3, 2006.
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