5,070 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
On the Construction of Splitting Methods by Stabilizing Corrections with Runge-Kutta Pairs
In this technical note a general procedure is described to construct
internally consistent splitting methods for the numerical solution of
differential equations, starting from matching pairs of explicit and diagonally
implicit Runge-Kutta methods. The procedure will be applied to suitable
second-order pairs, and we will consider methods with or without a mass
conserving finishing stage. For these splitting methods, the linear stability
properties are studied and numerical test results are presented.Comment: 18 page
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
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