Abstract
In this paper, trigonometrically fitted multi-step Runge-Kutta (TFMSRK) methods for the numerical integration of oscillatory initial value problems are proposed and studied. TFMSRK methods inherit the frame of multi-step Runge-Kutta (MSRK) methods and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of \(\{\exp (\mathrm {i}wt),\exp (-\mathrm {i}wt)\},\) or equivalently the set \(\{\cos (wt),\sin (wt)\}\), where w represents an approximation of the main frequency of the problem. The general order conditions are given and four new explicit TFMSRK methods with order three and four, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.
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References
Petzold, L.R., Jay, L.O., Yen, J.: Numerical solution of highly oscillatory ordinary differential equations. Acta Numer. 6, 437–483 (1997)
Van der Houwen, P.J., Sommeijer, B.P.: Diagonally implicit Runge-Kutta-Nyström methods for oscillatory problems. SIAM J. Numer. Anal. 26, 414–429 (1989)
Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations I: nonstiff problems, 2nd ed. Springer-Verlag, Berlin (2002)
Hairer, E., Wanner, G.: Solving ordinary differential equations II: stiff and differential-algebraic problems, 2nd ed. Springer-Verlag, Berlin (2002)
Ixaru, L.G.: Exponential Fitting, Kluwer Academic Publishers, Dordrecht, Boston. London (2004)
Simos, T.E.: A family of fifth algebraic order trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation. Comput. Mater. Sci. 34, 342–354 (2005)
Simos, T.E., Aguiar, J.V.: A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation. J. Math. Chem. 30, 121–131 (2001)
Franco, J.M.: Runge-Kutta methods adapted to the numerical integration of oscillatory problems. Numer. Math. Appl. 50, 427–443 (2004)
Van, de Vyver, H.: An embedded exponentially fitted Runge-Kutta-Nyström method for the numerical solution of orbital problems. New Astron. 11, 577–587 (2006)
Gautschi, W.: Numerical integration of ordinary differential equation based on trigonometric polynomials. Numer. Math. 3, 381–397 (1961)
Lyche, T.: Chebyshevian multistep methods for ordinary differential equations. Numer. Math. 19, 65–75 (1972)
Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially fitted Runge-Kutta methods. Comput. Phys. Comm. 123, 7–15 (1999)
Vanden Berghe, G., De Meyer, H., Van Daele, M., Van Hecke, T.: Exponentially fitted explicit Runge-Kutta methods. J. Comput. Appl. Math. 125, 107–115 (2000)
Paternoster, B.: Runge-Kutta(-Nystrom) methods for ODEs with periodic solutions based on trigonometric polynomials. Appl. Numer. Math. 28, 401–412 (1998)
Burrage, K.: High order algebraically stable multistep Runge-Kutta methods. SIAM J. Numer. Anal. 24, 106–115 (1987)
Burrage, K.: Order properties of implicit multivalue methods for ordinary 245 differential equations. IMA J. Numer. Anal 8, 43–69 (1988)
Butcher, J.C.: Numerical methods for ordinary differential equations, 2nd edn. Wiley, Chichester (2008)
Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge-Kutta methods for ordinary differential equations. SIAM J. Numer. Anal. 32, 1390–1427 (1995)
Jackiewicz, Z., Tracogna, S.: Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations. Numer. Algor. 12, 347–368 (1996)
Bartoszewski, Z., Jackiewicz, Z.: Construction of two-step Runge-Kutta methods of high order for ordinary differential equations. Numer. Algor. 18, 51–70 (1998)
Hairer, E., Wanner, G.: Order conditions for general two-step Runge-Kutta Methods. SIAM J. Numer. Anal. 34, 2087–2089 (1997)
Butcher, J.C., Tracogna, S.: Order conditions for two-step Runge-Kutta methods. Appl. Numer. Math. 24, 351–364 (1997)
Jackiewicz, Z.: General linear methods for ordinary differential equations. Wiley, New York (2009)
Conte, D., D’Ambrosio, R., Jackiewicz, Z., Paternoster, B.: Numerical search for algebraically stable two-step almost collocation methods. J. Comput. Appl. Math. 239, 304–321 (2013)
D’Ambrosio, R., Paternoster, B.: Two-step modified collocation methods with structured coefficient matrices. Appl. Numer. Math. 62, 1325–1334 (2012)
D’Ambrosio, R., Esposito, E., Paternoster, B.: Exponentially fitted two-step Runge-Kutta methods: Construction and parameter selection. Appl. Math. Comput. 218, 7468–7480 (2012)
Franco, J.M.: Exponentially fitted explicit Runge-Kutta-Nyström methods. J. Comput. Appl. Math. 167, 1–19 (2004)
Van, de Vyver, H.: : Scheifele two-step methods for perturbed oscillators. J. Comput. Appl. Math. 224, 415–432 (2009)
Franco, J.M.: A class of explicit two-step hybrid methods for second-order IVPs. J. Comput. Appl. Math. 187, 41–57 (2006)
Franco, J.M.: Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)
Ramos, H., Vigo-Aguiar, J.: On the frequency choice in trigonometrically fitted methods. Appl. Math. Lett. 23, 1378–1381 (2010)
Vigo-Aguiar, J., Ramos, H.: On the choice of the frequency in trigonometrically-fitted methods for periodic problems. J. Comput. Appl. Math. 277, 94–105 (2015)
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The research was supported in part by the Natural Science Foundation of China under Grant No: 11401164, by the Hebei Natural Science Foundation of China under Grant No: A2014205136, by the Natural Science Foundation of China under Grant No: 11201113 and by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant No: 20121303120001.
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Li, J. Trigonometrically fitted multi-step Runge-Kutta methods for solving oscillatory initial value problems. Numer Algor 76, 237–258 (2017). https://doi.org/10.1007/s11075-016-0252-2
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DOI: https://doi.org/10.1007/s11075-016-0252-2