22 research outputs found
Extrapolation-based implicit-explicit general linear methods
For many systems of differential equations modeling problems in science and
engineering, there are natural splittings of the right hand side into two
parts, one non-stiff or mildly stiff, and the other one stiff. For such systems
implicit-explicit (IMEX) integration combines an explicit scheme for the
non-stiff part with an implicit scheme for the stiff part.
In a recent series of papers two of the authors (Sandu and Zhang) have
developed IMEX GLMs, a family of implicit-explicit schemes based on general
linear methods. It has been shown that, due to their high stage order, IMEX
GLMs require no additional coupling order conditions, and are not marred by
order reduction.
This work develops a new extrapolation-based approach to construct practical
IMEX GLM pairs of high order. We look for methods with large absolute stability
region, assuming that the implicit part of the method is A- or L-stable. We
provide examples of IMEX GLMs with optimal stability properties. Their
application to a two dimensional test problem confirms the theoretical
findings
Extrapolation-Based Implicit-Explicit Peer Methods with Optimised Stability Regions
In this paper we investigate a new class of implicit-explicit (IMEX) two-step
methods of Peer type for systems of ordinary differential equations with both
non-stiff and stiff parts included in the source term. An extrapolation
approach based on already computed stage values is applied to construct IMEX
methods with favourable stability properties. Optimised IMEX-Peer methods of
order p = 2, 3, 4, are given as result of a search algorithm carefully designed
to balance the size of the stability regions and the extrapolation errors.
Numerical experiments and a comparison to other implicit-explicit methods are
included.Comment: 21 pages, 6 figure
Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part
In this paper, we extend the implicit-explicit (IMEX) methods of Peer type
recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203--215, 2017]
to a broader class of two-step methods that allow the construction of
super-convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes
combine the necessary stability of implicit and low computational costs of
explicit methods to efficiently solve systems of ordinary differential
equations with both stiff and non-stiff parts included in the source term. To
construct super-convergent IMEX-Peer methods with favourable stability
properties, we derive necessary and sufficient conditions on the coefficient
matrices and apply an extrapolation approach based on already computed stage
values. Optimised super-convergent IMEX-Peer methods of order s+1 for s=2,3,4
stages are given as result of a search algorithm carefully designed to balance
the size of the stability regions and the extrapolation errors. Numerical
experiments and a comparison to other IMEX-Peer methods are included.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1610.0051
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
Transformed implicit-explicit DIMSIMs with strong stability preserving explicit part
For many systems of differential equations modeling problems in science and
engineering, there are often natural splittings of the right hand side into two
parts, one of which is non-stiff or mildly stiff, and the other part is stiff.
Such systems can be efficiently treated by a class of implicit-explicit (IMEX)
diagonally implicit multistage integration methods (DIMSIMs), where the stiff
part is integrated by implicit formula, and the non-stiff part is integrated by
an explicit formula. We will construct methods where the explicit part has
strong stability preserving (SSP) property, and the implicit part of the method
is -, or -stable. We will also investigate stability of these methods
when the implicit and explicit parts interact with each other. To be more
precise, we will monitor the size of the region of absolute stability of the
IMEX scheme, assuming that the implicit part of the method is -, or
-stable. Finally we furnish examples of SSP IMEX DIMSIMs up to the order
four with good stability properties
Extrapolation-Based Implicit-Explicit Peer Methods with Optimised Stability Regions
In this paper we investigate an new class of implicit-explicit two-step
methods of Peer type for systems of ordinary differential equations with both
non-stiff and stiff parts included in the source term. An extrapolation
approach based on already computed stage values with equally high consistency
order is applied to construct such methods with strong stability properties.
Optimised implicit-explicit Peer methods of order p=2,3,4, are given as result
of a search algorithm carefully designed to balance the size of the stability
regions and the extrapolation errors. Numerical experiments and a comparison to
other implicit-explicit methods are included