American Institute of Mathematical Sciences

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January  2014, 19(1): 281-298. doi: 10.3934/dcdsb.2014.19.281

Dirichlet series for dynamical systems of first-order ordinary differential equations

Bin Wang 1, and  Arieh Iserles 2,

1. 

School of Mathematics & Physics, Qingdao University of Science & Technology, Qingdao 266061, China
2. 

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, United Kingdom

Received  August 2012 Revised  October 2013 Published  December 2013

In this paper, inspired by the work by A. Iserles and G. Söderlind [Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), pp. 97-112], we present comprehensive discussion on Dirichlet series for dynamical systems of first-order ordinary differential equations (ODEs). We first derive the scheme of Dirichlet approximation for scalar dynamical systems and present the bounds on the terms of Dirichlet series. The global error and the right choice of a term in Dirichlet series are analysed and two numerical experiments are carried out to demonstrate the efficiency of Dirichlet approximation. Then we consider applying Dirichlet series to multivariate dynamical systems and present a new scheme of Dirichlet approximation for such systems. Some discussion and a numerical experiment are accordingly carried out for the new Dirichlet approximation. Compared with routine time-stepping algorithms, Dirichlet series does not need time stepping and yields a continuous solution that is equally valid along an interval, which is significant for obtaining long-time numerical solution. As a result of the special nature of Dirichlet series, the Dirichlet approximation delivers considerable information on dynamical systems of first-order ODEs and provides a novel and effective approach to numerical solutions of these dynamical systems.
Keywords: dynamical systems, stationary points, global bounds., Dirichlet series, first-order ordinary differential equations.
Mathematics Subject Classification: Primary: 37M10, 37N30; Secondary: 65L05, 65L7.
Citation: Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281
References:
[1]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley and Sons, Ltd 2008. doi: 10.1002/9780470753767.  Google Scholar

[2]

Y. Fang, Y. Song and X. Wu, New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems, Phys. Lett. A, 372 (2008), 6551-6559. doi: 10.1016/j.physleta.2008.09.014.  Google Scholar

[3]

A. B. González, P. Martín and J. M. Farto, A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators, Numer. Math., 82 (1999), 635-646. doi: 10.1007/s002110050434.  Google Scholar

[4]

E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numer., 12 (2003), 399-450. doi: 10.1017/S0962492902000144.  Google Scholar

[5]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, 1993.  Google Scholar

[6]

G. H. Hardy and M. Riesz, The general theory of Dirichlet series, Cambridge Tracts in Mathematics and Mathematical Physics, 18 Stechert-Hafner, Inc., New York (1964).  Google Scholar

[7]

M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilineal parabolic problems, SIAM J. Numer. Anal., 43 (2005), 1069-1090. doi: 10.1137/040611434.  Google Scholar

[8]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd ed., Cambridge University Press, Cambridge, 2008.  Google Scholar

[9]

A. Iserles, G. P. Ramaswami and M. Sofroniou, Runge-Kutta methods for quadratic ordinary differential equations, BIT, 38 (1998), 315-346. doi: 10.1007/BF02512370.  Google Scholar

[10]

A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), 97-112. doi: 10.1006/jcom.1993.1007.  Google Scholar

[11]

A. Iserles and A. Zanna, Preserving algebraic invariants with Runge-Kutta methods, J. Comp. Appl. Maths., 125 (2000), 69-81. doi: 10.1016/S0377-0427(00)00459-3.  Google Scholar

[12]

S. Mandelbrojt, Dirichlet Series: Principles and Methods, D. Reidel Publishing Company, Dordrecht, Holland, 1972.  Google Scholar

[13]

L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer-Verlag, New York, 2001.  Google Scholar

[14]

R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete, Pearson Prentice Hall, New Jersey, 2004.  Google Scholar

[15]

J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems, BIT, 28 (1988), 877-883. doi: 10.1007/BF01954907.  Google Scholar

[16]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-642-97149-5.  Google Scholar

show all references

References:
[1]

J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd ed., John Wiley and Sons, Ltd 2008. doi: 10.1002/9780470753767.  Google Scholar

[2]

Y. Fang, Y. Song and X. Wu, New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems, Phys. Lett. A, 372 (2008), 6551-6559. doi: 10.1016/j.physleta.2008.09.014.  Google Scholar

[3]

A. B. González, P. Martín and J. M. Farto, A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators, Numer. Math., 82 (1999), 635-646. doi: 10.1007/s002110050434.  Google Scholar

[4]

E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method, Acta Numer., 12 (2003), 399-450. doi: 10.1017/S0962492902000144.  Google Scholar

[5]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, 1993.  Google Scholar

[6]

G. H. Hardy and M. Riesz, The general theory of Dirichlet series, Cambridge Tracts in Mathematics and Mathematical Physics, 18 Stechert-Hafner, Inc., New York (1964).  Google Scholar

[7]

M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilineal parabolic problems, SIAM J. Numer. Anal., 43 (2005), 1069-1090. doi: 10.1137/040611434.  Google Scholar

[8]

A. Iserles, A First Course in the Numerical Analysis of Differential Equations, 2nd ed., Cambridge University Press, Cambridge, 2008.  Google Scholar

[9]

A. Iserles, G. P. Ramaswami and M. Sofroniou, Runge-Kutta methods for quadratic ordinary differential equations, BIT, 38 (1998), 315-346. doi: 10.1007/BF02512370.  Google Scholar

[10]

A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), 97-112. doi: 10.1006/jcom.1993.1007.  Google Scholar

[11]

A. Iserles and A. Zanna, Preserving algebraic invariants with Runge-Kutta methods, J. Comp. Appl. Maths., 125 (2000), 69-81. doi: 10.1016/S0377-0427(00)00459-3.  Google Scholar

[12]

S. Mandelbrojt, Dirichlet Series: Principles and Methods, D. Reidel Publishing Company, Dordrecht, Holland, 1972.  Google Scholar

[13]

L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer-Verlag, New York, 2001.  Google Scholar

[14]

R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete, Pearson Prentice Hall, New Jersey, 2004.  Google Scholar

[15]

J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems, BIT, 28 (1988), 877-883. doi: 10.1007/BF01954907.  Google Scholar

[16]

F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-3-642-97149-5.  Google Scholar

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