Abstract
The purpose of this paper is to provide a parallel acceleration of peer methods for the numerical solution of systems of Ordinary Differential Equations (ODEs) arising from the space discretization of Partial Differential Equations (PDEs) modeling the growth of vegetation in semi-arid climatic zones. The parallel algorithm is implemented by using the CUDA environment for Graphics Processing Units (GPUs) architectures. Numerical experiments, showing the performance gain of the proposed strategy, are provided.
The authors are members of the GNCS group. This work is supported by GNCS-INDAM project and by the Italian Ministry of University and Research (MUR), through the PRIN 2020 project (No. 2020JLWP23) “Integrated Mathematical Approaches to Socio-Epidemiological Dynamics” (CUP: E15F21005420006) and the PRIN 2017 project (No. 2017JYCLSF) “Structure preserving approximation of evolutionary problems”.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Butcher, J.C.: Implicit Runge-Kutta processes. Math. Comp. 18, 50–64 (1964)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, Chichester (2008)
Butcher, J.C.: General linear methods. Acta Numer. 15, 157–256 (2006)
Calvo, M.P., Gerisch, A.: Linearly implicit Runge-Kutta methods and approximate matrix factorization. Appl. Math. 53(2–4), 183–200 (2005)
Conte, D., D’Ambrosio, R., Paternoster, B.: GPU-acceleration of waveform relaxation methods for large differential systems. Numer. Algorithms 71(2), 293–310 (2015). https://doi.org/10.1007/s11075-015-9993-6
Conte, D., Paternoster, B.: Parallel methods for weakly singular Volterra integral equations on GPUs. Appl. Numer. Math. 114, 30–37 (2016)
Cuomo, S., De Michele, P., Galletti, A., Marcellino, L.: A GPU-parallel algorithm for ECG signal denoising based on the NLM method. In: 2016 30th International Conference on Advanced Information Networking and Applications Workshops (WAINA), pp. 35–39, March 2016
Conte, D., D’Ambrosio, R., Pagano, G., Paternoster, B.: Jacobian-dependent vs. Jacobian-free discretizations for nonlinear differential problems. Comput. Appl. Math. 39(3), 1–12 (2020). https://doi.org/10.1007/s40314-020-01200-z
Conte, D., Mohammadi, F., Moradi, L., Paternoster, B.: Exponentially fitted two-step peer methods for oscillatory problems. Comput. Appl. Math. 39(3), 1–19 (2020). https://doi.org/10.1007/s40314-020-01202-x
Conte, D., Pagano, G., Paternoster, B.: Jacobian-dependent two-stage peer method for ordinary differential equations. In: Gervasi, O., et al. (eds.) ICCSA 2021, Part I. LNCS, vol. 12949, pp. 309–324. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-86653-2_23
Conte, D., Pagano, G., Paternoster, B.: Two-step peer methods with equation-dependent coefficients. Comput. Appl. Math. 41(4), 140 (2022)
Eigentler, L., Sherratt, J.A.: Metastability as a coexistence mechanism in a model for dryland vegetation patterns. Bull. Math. Biol. 81, 2290–2322 (2019). https://doi.org/10.1007/s11538-019-00606-z
Schmitt, B.A., Weiner, R.: Parallel start for explicit parallel two-step peer methods. Numer. Algorithms 53(2), 363–381 (2010). https://doi.org/10.1007/s11075-009-9267-2
Schmitt, B.A., Weiner, R., Jebens, S.: Parameter optimization for explicit parallel peer two-step methods. Appl. Numer. Math. 59(3–4), 769–782 (2009)
Schmitt, B.A., Weiner, R., Podhaisky, H.: Multi-implicit peer two-step W-methods for parallel time integration. BIT Numer. Math. 45(1), 197–217 (2005). https://doi.org/10.1007/s10543-005-2635-y
Schmitt, B.A., Weiner, R., Erdmann, K.: Implicit parallel peer methods for stiff initial value problems. Appl. Numer. Math. 53(2–4), 457–470 (2005)
Weiner, R., Schmitt, B.A., Podhaisky, H.: Parallel “Peer” two-step W-methods and their application to MOL-systems. Appl. Numer. Math., 48(3–4), 425–439 (2004)
Schmitt, B.A., Weiner, R.: Parallel two-step W-methods with peer variables. SIAM J. Numer. Anal. 42(1), 265–282 (2004)
De Luca, P., Galletti, A., Marcellino, L.: A Gaussian recursive filter parallel implementation with overlapping. In: 2019 15th International Conference on Signal-Image Technology and Internet-Based systems (SITIS), pp. 641–648 (2019)
De Luca, P., Galletti, A., Giunta, G., Marcellino, L.: Accelerated Gaussian convolution in a data assimilation scenario. In: Krzhizhanovskaya, V.V., et al. (eds.) ICCS 2020, Part VI. LNCS, vol. 12142, pp. 199–211. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50433-5_16
De Luca, P., Galletti, A., Ghehsareh, H.R., Marcellino, L., Raei, M.: A GPU-CUDA framework for solving a two-dimensional inverse anomalous diffusion problem. In: Foster, I., Joubert, G.R., Kučera, L., Nagel, W.E., Peters, F. (eds.) Parallel Computing: Technology Trends, Advances in Parallel Computing, vol. 36, pp. 311–320 (2020)
De Luca, P., Galletti, A., Giunta, G., Marcellino, L.: Recursive filter based GPU algorithms in a data assimilation scenario. J. Comput. Sci. 53, 101339 (2021)
Jones, S.: Introduction to dynamic parallelism. In: GPU Technology Conference Presentation, vol. 338 (2012)
Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin (1993). https://doi.org/10.1007/978-3-540-78862-1
Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, Berlin (2003). https://doi.org/10.1007/978-3-662-09017-6
Ixaru, L.G.: Runge-Kutta methods with equation dependent coefficients. Comput. Phys. Commun. 183(1), 63–69 (2012)
Jebens, S., Weiner, R., Podhaisky, H., Schmitt, B.: Explicit multi-step peer methods for special second-order differential equations. Appl. Math. Comput. 202(2), 803–813 (2008)
Klinge, M., Weiner, R., Podhaisky, H.: Optimally zero stable explicit peer methods with variable nodes. BIT Numer. Math. 58(2), 331–345 (2017). https://doi.org/10.1007/s10543-017-0691-8
Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5(4), 329–330 (1963)
Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50, 405–418 (1986). https://doi.org/10.1007/BF01396661
Weiner, R., Biermann, K., Schmitt, B., Podhaisky, H.: Explicit two-step peer methods. Comput. Math. Appl. 55(4), 609–619 (2008)
Acknowledgements
The authors are members of the GNCS group. This work is supported by GNCS-INDAM project and by the Italian Ministry of University and Research (MUR), through the PRIN 2020 project (No. 2020JLWP23) “Integrated Mathematical Approaches to Socio-Epidemiological Dynamics” (CUP: E15F21005420006) and the PRIN 2017 project (No. 2017JYCLSF) “Structure preserving approximation of evolutionary problems”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Conte, D. et al. (2022). First Experiences on Parallelizing Peer Methods for Numerical Solution of a Vegetation Model. In: Gervasi, O., Murgante, B., Hendrix, E.M.T., Taniar, D., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2022. ICCSA 2022. Lecture Notes in Computer Science, vol 13376. Springer, Cham. https://doi.org/10.1007/978-3-031-10450-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-031-10450-3_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-10449-7
Online ISBN: 978-3-031-10450-3
eBook Packages: Computer ScienceComputer Science (R0)