##### Report

on the usage of the resources of the HybriLIT heterogeneous platform

**Ivan Georgiev Hristov, Radoslava Danailova Hristova**

Faculty of Mathematics and Informatics, Sofia University, Bulgaria

e-mails: christov_ivan@abv.bg*, *radoslava@fmi.uni-sofia.bg

##### 1. Title of the task

Numerical investigation of a class of combined Explicit-Implicit Taylor series methods of various order of accuracy for solving Hamiltonian systems.

##### 2. Team members

N. Dimova, I. G. Hristov, R. D. Hristova, I. V. Puzynin, T. P. Puzynina, Z. A. Sharipov, N. G. Shegunov, Z. K. Tukhliev

##### 3. Abstract of your task with an indication of the PTP JINR theme

We investigate a class of combined Explicit-Implicit Taylor series methods of various order of accuracy for solving Hamiltonian systems. The purpose of the investigation is to confirm our expectations that these methods behave as symplectic ones in terms of energy conservation, and that in some cases they may overmatch the standard second order Verlet method. In addition we test an OpenMP approach for computing multiple independent trajectories using our methods. The results are very promising. We achieve a significant speedup up to, when we use the whole resource of one computational CPU node in the HybriLIT education and testing polygon.

*The work was made in the framework of the **PTP JINR theme** “Methods, Algorithms and Software for Modeling Physical Systems, Mathematical Processing and Analysis of Experimental Data” **05-6-1119-2014/2019*

##### 4. Description of the obtained results during the year

- The numerical tests show excellent long-time behavior of the total energy for the combined Explicit-Implicit Taylor series methods. This means that with respect to the energy conservation our methods behave as symplectic ones
*(see Figure 1).* - The average number of iterations for the implicit step depends on the step size and converges to 1 as tends to 0. For sufficiently small step size the work for one step is fixed and small and our methods behave as explicit ones
*(see the Table 1).* - For problems, requiring high accuracy, the considered methods show substantially better performance than the standard second order Verlet method
*(see the Figure 2).* - The results for computing multiple independent trajectories with our methods using OpenMP parallel technology show significant speedup up to 37, when we use the whole resource of one computational CPU node in the HybriLIT education and testing polygon
*(see Figure 3).*

** **

##### 5. Description of the resources used in the course of work: number of CPU cores, number of GPU (or CUDA cores), disk space amount. Please also list the used software packages

All of the numerical tests are performed on the HybriLIT education and testing polygon. Our computations are performed on one computational CPU-node consisting of 2 x Intel(R) Xeon(R) Processor E5-2695v3 (28 cores, 56 hardware threads). For compilations the Intel (R) Fortran Compiler 18.0 is used. We did not used any special software packages. The disk space amount was insignificant.

##### 6. List of publications containing acknowledgments for using the resource of the HybriLIT platform

We accept that all results provided in the report can be published at the HybriLIT website.

- N. Dimova, I. G. Hristov, R. D. Hristova, I. V. Puzynin, T. P. Puzynina, Z. A. Sharipov, N. G. Shegunov, Z. K. Tukhliev, Combined explicit-implicit taylor series methods CEUR Workshop Proceedings, GRID 2018 International Conference “Distributed Computing and Grid-technologies in Science and Education” 2018, volume 2267, 2018, pp.544-548, ISSN (online):1613-