JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 937858 10.1155/2013/937858 937858 Research Article A New Trigonometrically Fitted Two-Derivative Runge-Kutta Method for the Numerical Solution of the Schrödinger Equation and Related Problems Zhang Yanwei 1 Che Haitao 2 http://orcid.org/0000-0002-2159-7833 Fang Yonglei 1 You Xiong 3 Vigo-Aguiar Jesus 1 Department of Mathematics and Information Science Zaozhuang University Zaozhuang 277160 China uzz.edu.cn 2 School of Mathematics and Information Science Weifang University Weifang, Shandong 261061 China wfmc.edu.cn 3 Department of Applied Mathematics Nanjing Agricultural University Nanjing 210095 China njau.edu.cn 2013 3 12 2013 2013 04 07 2013 28 09 2013 04 10 2013 2013 Copyright © 2013 Yanwei Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new trigonometrically fitted fifth-order two-derivative Runge-Kutta method with variable nodes is developed for the numerical solution of the radial Schrödinger equation and related oscillatory problems. Linear stability and phase properties of the new method are examined. Numerical results are reported to show the robustness and competence of the new method compared with some highly efficient methods in the recent literature.

1. Introduction

We are interested in initial value problems related to systems of first-order ordinary differential equations (ODEs) of the form (1)y=f(x,y),y(x0)=y0, whose solutions have a pronounced oscillatory character. Such problems often occur in many applied fields such as celestial mechanics, physical chemistry, quantum mechanics, and electronics. The investigation of the numerical solution of (1) has been the subject of extensive research activity during the last few decades . Such special optimized methods fall into two classes. The first class consists of the methods with constant coefficients. These methods can be applied to any problem with periodic solution. Methods of the second class have coefficients depending on the frequency of problem. When a good estimate of the frequency is known in advance one can use procedures such as exponential/trigonometric fitting or phase fitting. For example, Chen et al.  considered phase-fitted and amplification-fitted RK methods whose updates are also phase fitted and amplification fitted. Their methods are shown to be more efficient than the codes in the literature for some typical test problems and for the Lotka-Volterra system and a two-gene regulatory network in biology as well. You et al.  investigated the trigonometrically fitted Scheifele methods for oscillatory problems. A good theoretical foundation of the exponential fitting techniques for multistep methods was presented by Gautschi  and Lyche .

Recently, Simos et al.  introduced an exponentially fitted explicit Runge-Kutta method which integrates exactly the model problem  y=iωy. Van de Vyver  constructed a new fourth-order explicit Runge-Kutta method based on Simos’ approach  for the numerical integration of the Schrödinger equation. And another exponentially fitted Runge-Kutta method with four stages was constructed by Vanden Berghe et al.  which exactly integrates differential initial value problems whose solutions are linear combinations of functions of the form  {exp(ωx),exp(-ωx)}. More recently, Chan and Tsai  presented a family of two-derivative Runge-Kutta (TDRK) methods. An advantage of the two-derivative Runge-Kutta methods over classical Runge-Kutta methods is that they can reach higher order with fewer function evaluations.

Inspired by the approaches of Simos  and of Chan and Tsai , we will construct in this paper a new two-derivative Runge-Kutta method for the numerical integration of the Schrödinger equation and related oscillatory problems. The remainder of this paper is organized as follows: in Section 2, we introduce the scheme of two-derivative Runge-Kutta methods and present (up to fifth) order conditions for TDRK methods. In Section 3, we construct a new trigonometrically fitted TDRK method and analyze its linear stability and phase properties. In Section 4, some numerical examples are given to show the effectiveness and competence of our new method compared to the selected methods in the recent literature. Section 5 is devoted to conclusive remarks.

2. Two-Derivative Runge-Kutta Methods

We start with a special form of explicit two-derivative Runge-Kutta (TDRK) methods proposed by Chan and Tsai : (2)Yi=yn+cihf(xn,yn)+h2j=1i-1aijg(xn+cjh,Yj),11111111111111111111111111i=2,,s,  yn+1=yn+hf(xn,yn)+h2i=1sbig(xn+cih,Yi), where  g(x,y)=y′′(x):=(f(x,y)/x)+(f(x,y)/y)f(x,y). The coefficients of the scheme can be expressed by the Butcher tableau (3)cAbT=0c2a21csas1ass-1b1bs-1bs or simplify by  (c,A,b). This explicit TDRK method involves only one evaluation of the function  f  and  s  evaluations of the function  g  per step.

According to Chan and Tsai , the conditions for order up to five are listed as follows.

Order 2 requires (4)i=1sbi=12.

Order 3 requires, in addition, (5)i=2sbici=16.

Order 4 requires, in addition, (6)i=2sbici2=112.

Order 5 requires, in addition, (7)i=2sbici3=120,i=3sj=2i-1biaijcj=1120.

In practice, the following simplifying assumption is useful: (8)i=1saij=12ci2,fori=2,,s. Choosing  c3=4/5  and solving (4)–(8) yield a three-stage TDRK method of Chan and Tsai  which is given by the following tableau: (9)0000131180045-212542125054892825336

3. A Fifth-Order TDRK Method with Frequency-Dependent Coefficients

In order to adapt TDRK methods to the problem (1) whose solutions share an oscillatory feature with  ω  being an accurate estimate of the principal frequency, we allow the coefficients to depend on  ω. In particular, we consider in this paper the three-stage explicit TDRK method given by the following Butcher tableau: (10)0000c2(ν)a21(ν)00c3(ν)a31(ν)a32(ν)0b1(ν)b2(ν)b3(ν)

3.1. Construction of the New Method

If we require the TDRK method (10) to integrate exactly  exp(iωx); that is, (11)exp(iν)=cos(ν)+isin(ν)=1+iν-k1ν2-ik2ν31.+k3ν4+ik4ν5-k5ν6,ν=ωh, then we have (12)cos(ν)=1-k1ν2+k3ν4-k5ν6,sin(ν)=ν-k2ν3+k4ν5, where (13)k1=b1(ν)+b2(ν)+b3(ν),k2=b2(ν)c2(ν)+b3(ν)c3(ν),k3=b2(ν)a21(ν)+b3(ν)(a31(ν)+a32(ν)),k4=b3(ν)a32(ν)c2(ν),k5=b3(ν)a32(ν)a21(ν). Solving (4)–(6) for  s=3, the simplifying conditions (8) and (12), we obtain the coefficients expressed in terms of  c3(ν)(14)b1(ν)=(6(ν2-4)+c3(ν)(72-24ν2+ν4)1+(24-72c3(ν))cos(ν)(ν2-4)1+6(1-2c3(ν))νsin(ν))×(6c3(ν)M)-1,a32(ν)=c3(ν)(-6ν+ν3+6sin(ν))2N(3ν4M(ν2-4+4cos(ν)+νsin(ν))),b2(ν)=(2c3(ν)-1)ν2(ν3-6ν+6sin(ν))2(3MN),b3(ν)=(-4+ν2+4cos(ν)+νsin(ν))(c3(ν)N),c2(ν)=(24-12ν2+ν4-24cos(ν))(-12ν2+2ν4+12νsin(ν)),a21(ν)=c2(ν)22,a31(ν)=c3(ν)22-a32(ν), in which (15)M=24-12ν2+ν4-24cos(ν),N=-24-12(c3(ν)-1)ν2+(2c3(ν)-1)ν4+24cos(ν)+12c3(ν)νsin(ν). Assuming that  c3(ν)  is a constant and letting (16)b2(ν)c23(ν)+b3(ν)c3(ν)3=120+𝒪(ν2), we obtain  c3=4/5. It is easy to check that (17)b3(ν)a32(ν)c2(ν)=1120-ν25040+. The limit of the above equation as  ν0  verifies the second equation of (7). Thus we obtain a new TDRK method of order five given by (14) with  c3=4/5. We denote this method by TFTDRKV5.

For small values of  |ν|  the above formula is subject to heavy cancelations and in that case the following series expansions should be used: (18)b1(ν)=548+ν2896-11ν42257920-37ν61390878720+4759ν845565186867200+,b2(ν)=928-3ν25488+529ν464538880+4507ν6208718737920-74446597ν8191453523919257600+,b3(ν)=25336-25ν243904-515ν4154893312+1115ν6222633320448+2904025ν810210854609027072+,a21(ν)=118+ν21512+23ν47620480-ν648898080-7153ν819222813209600+,a31(ν)=-2125+13ν21750-277ν42205000+713ν6582120000-1269227ν8133491758400000+,a32(ν)=42125-13ν21750+277ν42205000-713ν6582120000+1269227ν8133491758400000+,c2(ν)=13+ν2504+ν4317520-47ν6586776960-629ν8961140660480+.

It can be seen that when  ν0, this new method reduces to the fifth-order TDRK method (9) of Chan and Tsai . The local truncation error of the above method is given by (19)L.T.E11=-h621600{y(6)(xn)-gy(xn,yn)y(4)(xn)111111111111+20gyy(xn,yn)f(xn,yn)y(3)(xn)111111111111+20gxy(xn,yn)y(3)(xn)}+𝒪(h7).

3.2. Stability and Phase Analysis of the New Method

In this section, we investigate the linear stability of the new method. We consider the following test equation: (20)y=iλy,λ>0. When applied to (20) the TDRK method (2) produces the difference equation (21)yn+1=M(iθ,ν)yn,θ=λh,i2=-1, where (22)M(iθ,ν)=(1-θ2b(ν)T(I+θ2A(ν))-1e)+i(×(I+θ2A(ν))-1c(ν)θ(1-θ2)b(ν)Thhhh×(I+θ2A(ν))-1c(ν)) with  I  the identity matrix and  e=(1,,1)T.

Definition 1.

For the TDRK method (2) with stability function  M(iθ,ν), the region in the  θ-ν  plane (23)Ω:={(θ,ν):|M(iθ,ν)|1} is called the imaginary stability region of the method. And any closed curve defined by  |M(iθ,ν)|=1  is a stability boundary of the method.

The imaginary stability region of the TFTDRKV5 is plotted in Figure 1.

Stability region of method TFTDRKV5.

Definition 2 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

For the TDRK method (2) with stability function  M(iθ,ν), the quantities (24)P~(θ,ν)=θ-arg(M(iθ,ν)),D(θ,ν)=1-|M(iθ,ν)| are called the phase-lag (dispersion) and amplification factor error (dissipation), respectively. If (25)P~(θ,ν)=cϕθq+1+𝒪(θq+3),D~(θ,ν)=cdθp+1+𝒪(θp+3), the method is said to have phase-lag order  q  and dissipation order  p, respectively.

Denoting the ratio  r=ν/θ=ω/λ, we obtain the following expressions for the phase-lag and the amplification of the method TFTDRKV5: (26)P~(θ,rθ)=-r2-15040θ7+𝒪(θ9),D~(θ,rθ)=r2-15760θ8+𝒪(θ10). Thus, the method TFTDRKV5 has a phase-lag of order six and a dissipation of order seven.

4. Numerical Results

In this section we carry out six numerical results to illustrate the performance of the new method. The criterion used in the numerical comparison is the decimal logarithm of the maximum global error (LOG10(ERROR)) versus the computational effort measured by the CPU seconds (CPU SECONDS) required by each problem. The methods used for comparison are listed in the following.

EFRK4: the fourth-order exponentially fitted Runge-Kutta method given by Vanden Berghe et al. in .

RK4V: the fourth-order optimized Runge-Kutta method given by Van de Vyver in .

RK5S: the fifth-order trigonometrically fitted Runge-Kutta method derived by Anastassi and Simos in .

RK4S: the fourth-order exponentially fitted Runge-Kutta method derived by Simos in .

EFRK5: the fifth-order trigonometrically fitted Runge-Kutta method derived by Sakas and Simos in .

TFTDRKV5: the fifth-order TFTDRK method with one evaluation of function  f  and three evaluations of function  g  per step derived in Section 3.1 of this paper.

Problem 1.

Consider the numerical integration of the Schrödinger equation (27)y′′(x)=(v(x)-E)y(x), with the well-known Woods-Saxon potential (28)v(x)=c0z(1-a(1-z)), where  z=(exp(a(x-b)+1))-1,c0=-50,a=5/3, and  b=7.  The domain of numerical integration is  [0,15]. It is appropriate to choose  ω  as follows (see ): (29)ω={50+E,x[0,6.5],E,x[6.5,15]. In this experiment we consider the resonance problem (E>0). The numerical results are compared with the analytical solution of the Woods-Saxon potential, rounded to six decimal places. In Figures 2, 3, 4, and 5, we plot the error of  |Eanalytical-Ecalculated|  versus the computational effort measured by CPU seconds required by each method for  Eanalytical=53.588872,163.215341,341.495874,989.701916, respectively.

Efficiency curves for Problem 1:  E=53.588872.

Efficiency curves for Problem 1:  E=163.215341.

Efficiency curves for Problem 1:  E=341.495874.

Efficiency curves for Problem 1:  E=989.701916.

Problem 2.

We consider the inhomogeneous equation (30)y′′(x)+100y(x)=99sin(x),  y(0)=1,y(0)=11, whose exact solution is (31)y(x)=cos(10x)+sin(10x)+sin(x). In this test we choose  ω=10. The numerical results given in Figure 6 are computed with the stepsizes  h=2-i-4,i=1,,4,  for all the methods on the interval  [0,1000].

Efficiency curves for Problem 2.

Problem 3.

We consider the linear problem studied in  (32)y′′+10000y=(10000-4x2)cos(x2)-2sin(x2),y(0)=1,y(0)=100, whose exact solution is (33)y(x)=sin(100x)+cos(x2). In this test we take  ω=100. The numerical results given in Figure 7 are computed with the stepsizes  h=2-9-i,i=1,,4,  for all the methods on the interval  [0,100].

Efficiency curves for Problem 3.

Problem 4.

We consider the following “almost periodic” orbit problem studied by Franco and Palacios : (34)z′′+z=ϵeiψx,z(0)=1,z(0)=i,z, or equivalently (35)u′′+u=ϵcos(ψx),u(0)=1,u(0)=0,v′′+v=ϵsin(ψx),v(0)=0,v(0)=1, where  ϵ=0.001  and  ψ=0.01.  The analytic solution to the problem is given by (36)u(x)=1-ϵ-ψ21-ψ2cos(x)+ϵ1-ψ2cos(ψx),v(x)=1-ϵψ-ψ21-ψ2sin(x)+ϵ1-ψ2sin(ψx).

In this test we take  ω=1. The numerical results given in Figure 8 are computed with the stepsizes  h=2-i,i=1,,4,  for all the methods on the interval  [0,1000].

Efficiency curves for Problem 4.

Problem 5.

We consider the two-dimensional problem studied in  (37)y′′(t)+(13-12-1213)y(t)=(f1(t)f2(t)),y(0)=(10),y(0)=(-48), where (38)f1(t)=9cos(2t)-12sin(2t),f2(t)=-12cos(2t)+9sin(2t). The analytical solution is (39)y(t)=(sin(t)-sin(5t)+cos(2t)sin(t)-sin(5t)+sin(2t)). In this test we choose  ω=5. The numerical results given in Figure 9 are computed with the stepsizes  h=2-2-i,i=1,,5,  for all the methods on the interval  [0,100].

Efficiency curves for Problem 5.

Problem 6.

We consider the Fermi-Pasta-Ulam problem studied in . This problem consists of a chain of springs, where nonlinear springs alternate with stiff harmonic springs. The problem can be described by a Hamiltonian system with Hamiltonian function (40)H=12i=1N(ui2+vi2)+ω22i=1Nvi2+14i=1N(ui+1-vi+1-ui-vi)4, where  u0=v0=uN+1=vN+1=0. This is equivalent to the nonlinear oscillatory problem of the form (41)ui′′=-Hpui,i=1,,N,vi′′+ω2vi=-Hpvi,i=1,,N, where (42)Hp=14i=1N(ui+1-vi+1-ui-vi)4. Since the analytic solution is not available, we study the numerical total energy  H  on the interval  [0,tend]. In this test we choose  N=3  with  ω=100  and  tend=10  and we take the initial values (43)u1(0)=1,u1(0)=1,v1=1ω,v1(0)=1, the remaining initial values being zero.

The numerical results given in Figure 10 are computed with the stepsizes h=2-8-i,i=1,,4, for all the methods.

In Figures 210, we can see that the new method TFTDTRK5V is the most efficient among the six methods we select for comparison.

Efficiency curves for Problem 6.

5. Conclusions

A new trigonometrically fitted two-derivative Runge-Kutta method of algebraic order five is derived in this paper. We also analyze the linear stability and phase properties of the new method. Numerical results are reported to show the robustness and competence of the new method compared with some highly efficient methods in the recent literature. To explain the superiority of the new method, we observe that, compared with RK methods of a specific number of stages, TDRK methods have the possibility of gaining a higher algebraic order than traditional RK methods with the same number of stages. We conclude that, for problems with oscillatory solutions, trigonometrically fitted TDRK methods are more accurate and more efficient than the adapted (exponentially or trigonometrically fitted) RK methods.

Acknowledgments

The authors are grateful to the editors and the anonymous referees for their constructive comments and valuable suggestions which have helped to improve the paper. This research was partially supported by NSFC (no. 11101357 and no. 11171155), the foundation of Shandong Outstanding Young Scientists Award Project (no. BS2010SF031), the foundation of Scientific Research Project of Shandong Universities (no. J13LI03), NSF of Shandong Province, China (no. ZR2011AL006), the Fundamental Research Fund for the Central Universities (no. Y0201100265), and the foundation of Scientific Research Project of Weifang (no. 20121103).

Ixaru L. G. Rizea M. A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies Computer Physics Communications 1980 19 1 23 27 2-s2.0-0002792006 Simos T. E. A family of fifth algebraic order trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation Computational Materials Science 2005 34 4 342 354 2-s2.0-24044512978 10.1016/j.commatsci.2005.01.007 van de Vyver H. An embedded phase-fitted modified Runge-Kutta method for the numerical integration of the radial Schrödinger equation Physics Letters A 2006 352 4-5 278 285 2-s2.0-33644975341 10.1016/j.physleta.2005.12.020 Simos T. E. An embedded Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schrödinger equation International Journal of Modern Physics C 2000 11 6 1115 1133 10.1142/S0129183100000973 MR1803090 ZBL0985.65083 Simos T. E. Aguiar J. V. A modified phase-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation Journal of Mathematical Chemistry 2001 30 1 121 131 10.1023/A:1013185619370 MR1883509 ZBL1003.65082 Kalogiratou Z. Monovasilis Th. Simos T. E. Computation of the eigenvalues of the Schrödinger equation by exponentially-fitted Runge-Kutta-Nyström methods Computer Physics Communications 2009 180 2 167 176 10.1016/j.cpc.2008.09.001 MR2579975 ZBL1198.81088 Kosti A. A. Anastassi Z. A. Simos T. E. Construction of an optimized explicit Runge-Kutta-Nyström method for the numerical solution of oscillatory initial value problems Computers & Mathematics with Applications 2011 61 11 3381 3390 10.1016/j.camwa.2011.04.046 MR2802004 ZBL1222.65066 Kosti A. A. Anastassi Z. A. Simos T. E. An optimized explicit Runge-Kutta-Nyström method for the numerical solution of orbital and related periodical initial value problems Computer Physics Communications 2012 183 3 470 479 10.1016/j.cpc.2011.11.002 MR2875136 ZBL1264.65111 Franco J. M. Runge-Kutta methods adapted to the numerical integration of oscillatory problems Applied Numerical Mathematics 2004 50 3-4 427 443 10.1016/j.apnum.2004.01.005 MR2074013 ZBL1057.65043 Van de Vyver H. An embedded exponentially fitted Runge-Kutta-Nyström method for the numerical solution of orbital problems New Astronomy 2006 11 8 577 587 2-s2.0-33744726845 10.1016/j.newast.2006.03.001 Hairer E. Nørsett S. P. Wanner G. Solving Ordinary Differential Equations. I. Nonstiff Problems 1993 8 2nd Berlin, Germany Springer xvi+528 Springer Series in Computational Mathematics MR1227985 Farto J. M. González A. B. Martín P. An algorithm for the systematic construction of solutions to perturbed problems Computer Physics Communications 1998 111 1–3 110 132 10.1016/S0010-4655(98)00037-X MR1631151 ZBL0931.65072 Ramos H. Vigo-Aguiar J. On the frequency choice in trigonometrically fitted methods Applied Mathematics Letters 2010 23 11 1378 1381 10.1016/j.aml.2010.07.003 MR2718515 ZBL1197.65082 Vanden Berghe G. Ixaru L. Gr. De Meyer H. Frequency determination and step-length control for exponentially-fitted Runge-Kutta methods Journal of Computational and Applied Mathematics 2001 132 1 95 105 10.1016/S0377-0427(00)00602-6 MR1834805 ZBL0991.65062 Vigo-Aguiar J. Ramos H. Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations Journal of Computational and Applied Mathematics 2003 158 1 187 211 10.1016/S0377-0427(03)00473-4 MR2017377 ZBL1042.65053 Vanden Berghe G. Van Daele M. Exponentially-fitted Obrechkoff methods for second-order differential equations Applied Numerical Mathematics 2009 59 3-4 815 829 10.1016/j.apnum.2008.03.018 MR2492293 ZBL1166.65352 Chen Z. You X. Shu X. Zhang M. A new family of phase-fitted and amplification-fitted Runge-Kutta type methods for oscillators Journal of Applied Mathematics 2012 2012 27 236281 MR2991586 ZBL1268.65091 You X. Zhang Y. Zhao J. Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators Computer Physics Communications 2011 182 7 1481 1490 10.1016/j.cpc.2011.04.001 MR2793296 ZBL1262.65077 Gautschi W. Numerical integration of ordinary differential equations based on trigonometric polynomials Numerische Mathematik 1961 3 381 397 MR0138200 10.1007/BF01386037 ZBL0163.39002 Lyche T. Chebyshevian multistep methods for ordinary differential equations Numerische Mathematik 1972 19 65 75 MR0303737 10.1007/BF01395931 ZBL0221.65123 Simos T. E. An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions Computer Physics Communications 1998 115 1 1 8 10.1016/S0010-4655(98)00088-5 MR1665072 ZBL1001.65080 Van de Vyver H. Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems Computer Physics Communications 2005 173 3 115 130 10.1016/j.cpc.2005.07.007 MR2187062 ZBL1196.65117 Vanden Berghe G. De Meyer H. Van Daele M. Van Hecke T. Exponentially fitted Runge-Kutta methods Journal of Computational and Applied Mathematics 2000 125 1-2 107 115 10.1016/S0377-0427(00)00462-3 MR1803185 ZBL0999.65065 Chan R. P. K. Tsai A. Y. J. On explicit two-derivative Runge-Kutta methods Numerical Algorithms 2010 53 2-3 171 194 MR3002485 10.1007/s11075-009-9349-1 ZBL1185.65122 van der Houwen P. J. Sommeijer B. P. Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions SIAM Journal on Numerical Analysis 1987 24 3 595 617 10.1137/0724041 MR888752 ZBL0624.65058 Anastassi Z. A. Simos T. E. Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation Journal of Mathematical Chemistry 2005 37 3 281 293 10.1007/s10910-004-1470-8 MR2138627 ZBL1085.65063 Sakas D. P. Simos T. E. A fifth algebraic order trigonometrically-fitted modified Runge-Kutta Zonneveld method for the numerical solution of orbital problems Mathematical and Computer Modelling 2005 42 7-8 903 920 10.1016/j.mcm.2005.09.018 MR2178519 ZBL1085.65061 Franco J. M. Palacios M. High-order P-stable multistep methods Journal of Computational and Applied Mathematics 1990 30 1 1 10 10.1016/0377-0427(90)90001-G MR1051797 ZBL0726.65091 Franco J. M. A class of explicit two-step hybrid methods for second-order IVPs Journal of Computational and Applied Mathematics 2006 187 1 41 57 10.1016/j.cam.2005.03.035 MR2178096 ZBL1082.65071 Galgani L. Giorgilli A. Martinoli A. Vanzini S. On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates Physica D 1992 59 4 334 348 10.1016/0167-2789(92)90074-W MR1192748 ZBL0775.70023