Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?
Abstract
:1. Introduction
2. Preliminaries
2.1. An Example as a Motivation
3. Standard Krylov Method
3.1. Description of Subroutine Phipm.m
3.2. Application to Our Problem
3.3. Suggestion to Improve the Result with Phipm.m
3.4. Modification of Subroutine Phipm.m
4. Purely Rational Krylov Method
Application to Our Problem
5. General Conclusions
- At least for the range of values in which we move in the example of Section 2.1, calculating (6) or (7) with a given required accuracy is likely to be cheaper when r increases, in spite of the fact that, apparently, more terms need to be calculated.
- In the case that having to solve several linear systems at each step with a same matrix is not a drawback for having chosen an exponential method to integrate a problem like (1), we recommend to approximate (6) or (7) by using (19) or (20) and a purely rational Krylov method to calculate the last term there.
- Whenever having to solve several linear systems at each step is a drawback for using exponential methods to integrate a problem like (1), we recommend using the modification of the adaptive subroutine phipm.m, which is described in Section 3.4, although the subroutine phipm.m applied as described in Section 3.3 also gives quite acceptable results, at least in our precise example.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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6.5e-01 | 5.5e-01 | 5.1e-01 | 4.4e-01 | |
1.4e+00 | 1.6e+00 | 1.8e+00 | 2.1e+00 |
4.9e-01 | 4.2e-01 | 3.9e-01 | |
8.1e-01 | 6.5e-01 | 5.8e-01 |
3.9e-01 | 3.4e-01 | 3.5e-01 | |
1.0e+00 | 5.9e-01 | 5.3e-01 |
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Cano, B.; Reguera, N. Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost? Mathematics 2021, 9, 1008. https://doi.org/10.3390/math9091008
Cano B, Reguera N. Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost? Mathematics. 2021; 9(9):1008. https://doi.org/10.3390/math9091008
Chicago/Turabian StyleCano, Begoña, and Nuria Reguera. 2021. "Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?" Mathematics 9, no. 9: 1008. https://doi.org/10.3390/math9091008
APA StyleCano, B., & Reguera, N. (2021). Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost? Mathematics, 9(9), 1008. https://doi.org/10.3390/math9091008