Abstract
In this paper, we present a general class of exponentially fitted two-step peer methods for the numerical integration of ordinary differential equations. The numerical scheme is constructed in order to exploit a-priori known information about the qualitative behaviour of the solution by adapting peer methods already known in literature. Examples of methods with 2 and 3 stages are provided. The effectiveness of this problem-oriented approach is shown through some numerical tests on well-known problems.
Funding: This work is supported by GNCS-INDAM.
References
[1] M. Calvo, J. M. Franco, J. I. Montijano, and L. Rández, Explicit Runge–Kutta methods for initial value problems with oscillating solutions, J. Comput. Appl. Math., 76 (1996) No. 1-2, 195–212.10.1016/S0377-0427(96)00103-3Search in Google Scholar
[2] M. Calvo, J. I. Montijano, L. Rández, and M. Van Daele, On the derivation of explicit two-step peer methods, Appl. Numer. Math., 61 (2010) No. 4, 395–409.10.1016/j.apnum.2010.11.004Search in Google Scholar
[3] M. Calvo, J. I. Montijano, L. Rández, and M. Van Daele, Exponentially fitted fifth-order two step peer explicit methods, AIP Conf. Proc., 1648 (2015), 150015-1–150015-4.10.1063/1.4912445Search in Google Scholar
[4] A. Cardone, L. Gr. Ixaru, and B. Paternoster, Exponential fitting direct quadrature methods for Volterra integral equations, Numer. Algorithms, 55 (2010), 467–480.10.1007/s11075-010-9365-1Search in Google Scholar
[5] A. Cardone, L. Gr. Ixaru, B. Paternoster, and G. Santomauro, Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution, Math. Comput. Simulat., 110(C) (2015), 125–143.10.1016/j.matcom.2013.10.005Search in Google Scholar
[6] D. Conte, E. Esposito, L. Gr. Ixaru, and B. Paternoster, Some new uses of the ηm(Z) functions, Comput. Phys. Commun., 181 (2010), 128–137.10.1016/j.cpc.2009.08.007Search in Google Scholar
[7] D. Conte, L. Gr. Ixaru, B. Paternoster, and G. Santomauro, Exponentially-fitted Gauss–Laguerre quadrature rule for integrals over an unbounded interval, J. Comput. Appl. Math., 255 (2014), 725–736.10.1016/j.cam.2013.06.040Search in Google Scholar
[8] D. Conte and B. Paternoster, Modified Gauss–Laguerre Exponential Fitting Based Formulae, J. Sci. Comp., 69 (2016), No. 1, 227–243.10.1007/s10915-016-0190-0Search in Google Scholar
[9] R. D’Ambrosio, E. Esposito, and B. Paternoster, Exponentially fitted two-step Runge–Kutta methods: construction and parameter selection, Appl. Math. Comput., 218 (2012), 7468–7480.10.1016/j.amc.2012.01.014Search in Google Scholar
[10] R. D’Ambrosio, E. Esposito, and B. Paternoster, Parameter estimation in exponentially fitted hybrid methods for second order ordinary differential problems. J. Math. Chem., 50 (2012), 155–168.10.1007/s10910-011-9903-7Search in Google Scholar
[11] R. D’Ambrosio, M. Moccaldi, B. Paternoster, and F. Rossi, On the employ of time series in the numerical treatment of differential equations modelling oscillatory phenomena, Communications in Computer and Information Science, 708 (2017), 179–187.10.1007/978-3-319-57711-1_16Search in Google Scholar
[12] R. D’Ambrosio, M. Moccaldi, and B. Paternoster, Adapted numerical methods for advection–reaction–diffusion problems generating periodic wavefronts, Comput. Math. Appl., 74 (2017), No. 5, 1029–1042.10.1016/j.camwa.2017.04.023Search in Google Scholar
[13] R. D’Ambrosio and B. Paternoster, Exponentially fitted singly diagonally implicit Runge–Kutta methods, J. Comput. Appl. Math., 263 (2014), 277–287.10.1016/j.cam.2013.12.022Search in Google Scholar
[14] R. D’Ambrosio and B. Paternoster, Numerical solution of a diffusion problem by exponentially fitted finite difference methods. SpringerPlus3 (2014), 425, 7p.10.1186/2193-1801-3-425Search in Google Scholar
[15] R. D’Ambrosio and B. Paternoster, Numerical solution of reaction–diffusion systems of λ-ω type by trigonometrically fitted methods, J. Comput. Appl. Math., 294 (2016), 436–445.10.1016/j.cam.2015.08.012Search in Google Scholar
[16] R. D’Ambrosio, B. Paternoster, and G. Santomauro, Revised exponentially fitted Runge–Kutta–Nyström methods, Appl. Math. Lett., 30 (2014), 56–60.10.1016/j.aml.2013.10.013Search in Google Scholar
[17] W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3 (1961), 381–397.10.1007/BF01386037Search in Google Scholar
[18] H. Hairer, C. Lubich, and G. Wanner, Geometrical Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, II ed., Springer, Berlin, 2006.Search in Google Scholar
[19] L. Gr. Ixaru, Operations on oscillatory functions, Comput. Phys. Commun., 105 (1997), 1–19.10.1016/S0010-4655(97)00067-2Search in Google Scholar
[20] L. Gr. Ixaru, Runge–Kutta method with equation dependent coefficients, Comput. Phys. Commun., 183 (2012), 63–69.10.1016/j.cpc.2011.08.017Search in Google Scholar
[21] L. Gr. Ixaru and G. Vanden Berghe, Exponential Fitting, Kluwer, Boston–Dordrecht–London, 2004.10.1007/978-1-4020-2100-8Search in Google Scholar
[22] J. K. Kim, R. Cools, and L. Gr. Ixaru, Extended quadrature rules for oscillatory integrands, Appl. Numer. Math., 46 (2003), 59–73.10.1016/S0168-9274(03)00009-6Search in Google Scholar
[23] J. K. Kim, R. Cools, and L. Gr. Ixaru, Quadrature rules using first derivatives for oscillatory integrands, J. Comput. Appl. Math., 140 (2002), 479–497.10.1016/S0377-0427(01)00483-6Search in Google Scholar
[24] G. Y. Kulikov and R. Weiner, Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation, J. Comput. Appl. Math., 233 (2010), 2351–2364.10.1016/j.cam.2009.10.020Search in Google Scholar
[25] J. I. Montijano, L. Rández, M. Van Daele, and M. Calvo, Functionally fitted explicit two step peer methods, J. Sci. Comput., 64 (2014), No. 3, 938–958.10.1007/s10915-014-9951-9Search in Google Scholar
[26] K. Ozawa, A functional fitting Runge–Kutta method with variable coefficients, Japan. J. Indust. Appl. Math., 18 (2001), 107–130.10.1007/BF03167357Search in Google Scholar
[27] B. Paternoster, Runge–Kutta(–Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials, Appl. Numer. Math., 28 (1998), 401–412.10.1016/S0168-9274(98)00056-7Search in Google Scholar
[28] B. A. Schmitt and R. Weiner, Parallel start for explicit parallel two-step peer methods, Numer. Algor., 53 (2010), 363–381.10.1007/s11075-009-9267-2Search in Google Scholar
[29] Schmitt, B. A. and Weiner, R., Parallel two-step W-methods with peer variables, SIAM J. Numer. Anal., 42 (2004), 265–282.10.1137/S0036142902411057Search in Google Scholar
[30] B. A. Schmitt, R. Weiner, and K. Erdmann, Implicit parallel peer methods for stiff initial value problems, Appl. Numer. Math., 53 (2005), No. 2-4, 457–470.10.1016/j.apnum.2004.08.019Search in Google Scholar
[31] B. A. Schmitt, R. Weiner, and S. Jebens, Parameter optimization for explicit parallel peer two-step methods, Appl. Numer. Math., 59 (2009), 769–782.10.1016/j.apnum.2008.03.013Search in Google Scholar
[32] B. A. Schmitt, R. Weiner, and E. Podhaisky, Multi-implicit peer two-step W-methods for parallel time integration, BIT, 45 (2005), 197–217.10.1007/s10543-005-2635-ySearch in Google Scholar
[33] T. E. Simos, A fourth algebraic order exponentially-fitted Runge–Kutta method for the numerical solution of the Schrödinger equation, IMA J. Numer. Anal., 21 (2001), 919–931.10.1093/imanum/21.4.919Search in Google Scholar
[34] T. E. Simos, An exponentially-fitted Runge–Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions. Comput. Phys. Comm., 115 (1998), 1–8.10.1016/S0010-4655(98)00088-5Search in Google Scholar
[35] G. Vanden Berghe, H. De Meyer, M. Van Daele, and T. Van Hecke, Exponentially fitted explicit Runge–Kutta methods, Comput. Phys. Commun., 123 (1999), 7–15.10.1016/S0010-4655(99)00365-3Search in Google Scholar
[36] M. Van Daele, G. Vanden Berghe, and H. Vande Vyver, Exponentially fitted quadrature rules of Gauss type for oscillatory integrands. Appl. Numer. Math., 53 (2005), 509–526.10.1016/j.apnum.2004.08.018Search in Google Scholar
[37] M. Van Daele, T. Van Hecke, G. Vanden Berghe, and H. De Meyer, Deferred correction with mono-implicit Runge–Kutta methods for first-order IVPs, Numerical methods for differential equations, J. Comput. Appl. Math., 111 (1999), No. 1-2, 37–47.10.1016/S0377-0427(99)00130-2Search in Google Scholar
[38] R. Weiner, K. Biermann, B. A. Schmitt, and H. Podhaisky, Explicit two-step peer methods, Comput. Math. Appl., 55 (2008), 609–619.10.1016/j.camwa.2007.04.026Search in Google Scholar
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