Applied Numerical Mathematics 38 (2001) 135–144 Explicit pseudo two-step RKN methods with stepsize control ✩ Nguyen Huu Cong Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam Abstract This paper is devoted to variable stepsize strategy implementations of a class of explicit pseudo two-step Runge– Kutta–Nyström methods of arbitrarily high order for solving nonstiff problems for systems of special secondorder differential equations The constant stepsize explicit pseudo two-step Runge–Kutta–Nyström methods are developed into variable stepsize ones and equipped with embedded formulas giving a cheap error estimate for stepsize control By two examples of widely-used test problems, a pseudo two-step Runge–Kutta–Nyström method of order implemented with variable stepsize strategy is shown to be much more efficient than parallel and sequential codes available in the literature With stringent error tolerances, this new explicit pseudo two-step Runge–Kutta–Nyström method is even superior to sequential codes in a sequential computer 2001 IMACS Published by Elsevier Science B.V All rights reserved Keywords: Runge–Kutta–Nyström methods; Two-step Runge–Kutta–Nyström methods; Embedded formulas; Parallelism Introduction The arrival of parallel computers influences the development of methods for the numerical solution of a nonstiff initial value problem (IVP) for systems of special second-order ordinary differential equations (ODEs) y (t) = f t, y(t) , y(t0 ) = y , y (t0 ) = y , y, f ∈ Rd (1.1) The most efficient numerical methods for solving this problem are the explicit Runge–Kutta–Nyström methods (RKN methods) In the literature, sequential explicit RKN methods up to order 11 can be found in, e.g., [12–17,19,23] In order to exploit the facilities of parallel computers, several classes of parallel explicit methods have been investigated, for example, in [2–7,9,10] A common challenge in the latter mentioned works is to reduce, for a given order of accuracy, the required number of effective sequential f -evaluations per step, using parallel processors In our previous work [10], we have considered a ✩ This work was partly supported by DAAD, N.R.P.F.S and QG-96-02 E-mail address: nhcong@ncst.ac.vn (N.H Cong) 0168-9274/01/$ – see front matter 2001 IMACS Published by Elsevier Science B.V All rights reserved PII: S - ( ) 0 - 136 N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 general class of explicit pseudo two-step RKN methods (EPTRKN methods) for solving problems of the form (1.1) A general s-stage (constant stepsize) EPTRKN method based on an s-dimensional collocation vector c = (c1 , , cs )T with distinct abscissas ci has the form Y n = e ⊗ y n + hc ⊗ y n + h2 (A ⊗ I )F (tn−1 e + hc, Y n−1 ), y n+1 = y n + hy n + h2 bT ⊗ I F (tn e + hc, Y n ), y n+1 = y n + h d T ⊗ I F (tn e + hc, Y n ), (1.2a) (1.2b) where Y n = (Y n,i ) and F (tn e + hc, Y n ) = (f (tn + ci h, Y n,i )), both are sd-dimensional vectors This method has been specified by the tableau A c O y n+1 bT y n+1 d T The (constant) s-dimensional vectors b and d are the parameters of the generating implicit RKN method, s × s matrix A is given by (see [10, Section 2.1]) A = P Q−1 , j +1 P = (pij ) = ci , j +1 Q = (qij ) = j (ci − 1)j −1 , (1.3) i = 1, , s, j = 1, , s The method (1.2) is of order p = min{p ∗ , s + 2} and stage order q = s, where p ∗ is the order of the generating implicit RKN method (cf [10, Theorems 2.1 and 2.2]) The number of f -evaluations per step equals s in a sequential implementation and equals in a parallel implementation using s processors This class of EPTRKN methods implemented with constant stepsize was shown to be very efficient for the solution of problems with stringent accuracy demand (cf [10, Section 3]) In the present work, we equip the EPTRKN methods with the ability to change the stepsize in an implementation with stepsize control Since the EPTRKN methods are of a two-step nature, we consider the method with (variable) parameters which are functions of stepsizes (see Section 2) For a practical error estimation used in a stepsize selection, an approach for constructing embedded formulas is discussed in Section Notice that for EPTRKN methods, embedded formulas are provided without additional f -evaluations Finally, in Section 4, we present numerical comparisons of a variable stepsize strategy EPTRKN method with the codes DOPRIN, ODEX2 and PIRKN currently available for two widely used test examples taken from the literature Variable stepsize EPTRKN methods It is well known that an efficient integration method must be able to change stepsizes Because EPTRKN methods are of a two-step nature, there is an additional difficulty in using these methods with variable stepsize mode There exist in principle two approaches for overcoming this difficulty (cf., e.g., [1, p 44; 20, p 397]): • interpolating past stage values; N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 137 • deriving methods with variable parameters The first approach using polynomial interpolation to reproduce the starting stage values for the new step involves with computational cost which increases as the dimension of the problem increases, while for the second approach, the computational cost is independent of the dimension of the problem For this reason, the variable parameter approach is more feasible and robust Thus, we are confined to the second approach and consider the EPTRKN method Y n = e ⊗ y n + hn c ⊗ y n + h2n (An ⊗ I )F (tn−1 e + hn−1 c, Y n−1 ), (2.1a) y n+1 = y n + hn y n + h2n bT ⊗ I F (tn e + hn c, Y n ), (2.1b) y n+1 = y n + hn d T ⊗ I F (tn e + hn c, Y n ), with variable stepsize hn = tn+1 − tn and variable parameter matrix An Here as in (1.2), F (tn−1 e + hn−1 c, Y n−1 ) = (f (tn−1 + ci hn−1 , Y n−1,i )) and F (tn e + hn c, Y n ) = (f (tn + ci hn , Y n,i )) The order and stage order of a variable stepsize EPTRKN method is defined in the same way as in the case of constant stepsize EPTRKN methods (cf [10, Definition 2.1]) The matrix An in the method (2.1) can be determined by order conditions as a matrix function of the stepsize ratios following [8] The (s + 1)order conditions can be derived by replacing Y n−1 , y n and Y n in (2.1a) with the exact solution values y(tn−1 e + hn−1 c), y(tn ) and y(tn e + hn c), respectively, that is y(tn e + hn c) − e ⊗ y(tn ) − hn c ⊗ y (tn ) − h2n (An ⊗ I )y (tn−1 e + hn−1 c) = O hs+2 n (2.2) Let us suppose that the stepsize ratio hn / hn−1 is bounded from above (i.e., hn / hn−1 Ω), then along the same lines of [10, Section 2.1], using Taylor expansions, we can expand the left-hand sides of (2.2) in powers of hn and obtain the order conditions for determining An given by C (j +1) = (j + 1)! hn hn−1 j −1 cj +1 − j (j + 1)An (c − e)j −1 = 0, j = 1, , s (2.3a) The condition (2.3a) can be written in the form (cf (1.3)) An Q − P diag 1, hn hn , , hn−1 hn−1 s−1 = O, (2.3b) which gives the explicit expression of An defined as An = P diag 1, hn hn , , hn−1 hn−1 s−1 Q−1 (2.3c) The following lemma can easily be deduced from (2.3c) Lemma 2.1 For the variable stepsize EPTRKN method (2.1), the variable parameter matrix An is uniformly bounded whenever the stepsize ratio hn / hn−1 is bounded from above For hn / hn−1 Ω, the principal error vector C (s+2) is also uniformly bounded Consequently, similarly to the order considerations for a general variable stepsize multistep method (cf., e.g., [20, p 401]), the relations (2.3) imply that locally Y (tn e + hn c) − Y n = O hs+2 n 138 N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 Along the lines of the proof of Theorem 2.1 and Theorem 2.2 in [10], we have that if the function f is Lipschitz continuous and if the condition of Lemma 2.1 is satisfied then at tn+1 y(tn+1 ) − y n+1 = O hp+1 + O hs+4 , n n + O hs+3 , y (tn+1 ) − y n+1 = O hp+1 n n where p is the order of the associated constant stepsize EPTRKN method Hence, the order and stage order of the variable stepsize EPTRKN method defined by (2.1) and (2.3c) is identical with those of the associated constant stepsize EPTRKN method (see [10, Theorem 2.2], also Section 1) Thus we have: Theorem 2.1 An s-stage variable stepsize EPTRKN method defined by (2.1) with variable matrix An defined by (2.3c) is of order p = s and of stage order q = s for any collocation vector c with distinct abscissas ci if hn / hn−1 is bounded from above It has stage order q = s + and order p = s + or p = s + if in addition the orthogonality relation x Pj (1) = 0, ξ j −1 Pj (x) := is satisfied for j = or j s (ξ − ci ) dξ, i=1 2, respectively Remark 2.1 The condition hn / hn−1 Ω is a reasonable assumption for a numerical code Remark 2.2 Zero-stability property of an EPTRKN method is independent of the method parameters (see [10, Section 2.2]) so that the variable stepsize EPTRKN methods are always stable Embedded EPTRKN methods With the aim to have a cheap error estimate for stepsize control in an implementation of EPTRKN methods, in parallel with the pth-order method (2.1), we consider a second pth-order EPTRKN method based on collocation vector c = (c1 , , cs˜ )T of the form Y n = e ⊗ y n + hn c ⊗ y n + h2n A ⊗ I F tn−1 e + hn−1 c, Y n−1 , T y n+1 = y n + hn y n + h2n b ⊗ I F tn e + hn c, Y n , (3.1) T y n+1 = y n + hn d ⊗ I F tn e + hn c, Y n , where, p > p, the vector c is a subvector of the vector c, i.e., {c1 , , cs˜ } ⊂ {c1 , , cs } By introducing new parameter vectors b = (b1 , , bs )T and d = (d1 , , ds )T which are defined according to if ci = cj , then bi = bj , di = dj , j = 1, , s, else bi = 0, di = 0, i = 1, , s, (3.2) we obtain an embedded formula without additional f -evaluations given by T y n+1 = y n + hn y n + h2n b ⊗ I F (tn e + hn c, Y n ), T y n+1 = y n + hn d ⊗ I F (tn e + hn c, Y n ) (3.3) N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 139 Theorem 3.1 If the function f is Lipschitz continuous, then the numerical approximations at tn+1 defined by (1.2b) and by (3.3) locally satisfy the order relation ˆ , y n+1 − y n+1 = O hp+1 n (3.4) ˆ y n+1 − y n+1 = O hp+1 n Proof As the EPTRKN method (3.1) has order p less than order p of the EPTRKN method (1.2), we may write ˆ + (y n+1 − y n+1 ), y n+1 − y n+1 = (y n+1 − y n+1 ) + (y n+1 − y n+1 ) = O hp+1 n ˆ + y n+1 − y n+1 y n+1 − y n+1 = y n+1 − y n+1 + y n+1 − y n+1 = O hp+1 n (3.5a) Since the function f is Lipschitz continuous, from the definition of the vectors b and d in (3.2) we have y n+1 − y n+1 = (y n − y n ) + O hsn˜+4 , (3.5b) y n+1 − y n+1 = (y n − y n ) + O hsn˜+3 The relations (3.5) then prove Theorem 3.1 ✷ Thus, for a practical error estimation used in a stepsize selection we have the embedded EPTRKN method given by (2.1a), (2.1b) and (3.3) which can be specified by the tableau An c O y n+1 bT y n+1 d T y n+1 b y n+1 d T T The local error estimate in various vector norms is then defined by (3.4) By this approach of constructing embedded EPTRKN methods, there exist several embedded formulas for an EPTRKN method The “lower order estimator” approach for embedded formulas described above does not give asymptotically correct error estimates However, for this approach no additional sequential f -evaluations are required and the resulting embedded pair can be implemented on only s processors Further, let us consider another approach of error estimates Since all EPTRKN methods have the same sequential cost of one f -evaluation per step, in a parallel environment, it is possible, at no additional sequential cost to compute asymptotically correct error estimates by using parallel processors For such a purpose we define the embedded method of the form (3.1), where p is sufficiently greater than p (with s sufficiently greater than s) Then, the following order relations , y n+1 − y n+1 = O hp+1 n y n+1 − y n+1 = O hp+1 n 140 N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 give an asymptotically correct error estimate of local order p + (order of the original EPTRKN method (2.1)) For this latter approach, the number of processors for implementing the embedded pair {(2.1), (3.1)} is s + s Thus, with respect to the former “lower order estimator” approach, s additional processors are needed The latter approach is also involved with constructing embedded pairs of EPTRKN methods with reasonable orders and stability properties This could be a subject of a later work In the section of numerical experiments below, we confine our consideration to the simple “lower order estimator” approach which is similar to that applied in other sequential and parallel codes used in the numerical comparisons and which does not require any additional processors for implementation Numerical experiments In this section we shall report the numerical results obtained by a new (parallel) EPTRKN method of orders 8, two sequential codes DOPRIN, ODEX2 and a parallel code PIRKN taken from the literature DOPRIN is a code based RKN pair 7(6) due to Dormand and Prince which can be found in the first edition of [20] ODEX2 is an extrapolation code for special second-order ODEs of the form (1.1) It uses variable order and variable stepsize and is recognized as being one of the most efficient sequential integrators for nonstiff problems of the form (1.1) (see [20, p 484]) PIRKN is a parallel code based on the PIRKN method of order 12 taken from [25] The eighth-order EPTRKN method is based on the collocation vector c8 = (0.057, 0.277, 0.584, 0.860, 1.000, 1.277, 1.584, 1.860)T , (4.1a) with an embedded formula of order based on c = (0.057, 0.277, 0.584, 0.860, 1.277, 1.584, 1.860)T (4.1b) Notice that the choice of the collocation vector in (4.1a) minimizes the principal error terms for some stage approximated values (cf [10, Theorem 2.4]) and gives slightly larger stability boundary No special effort has been made to optimize the parameters of the above method An optimal choice of the method parameters was beyond the scope of this work The stability interval of the EPTRKN method defined by (4.1a) is numerically calculated to be (−0.596, 0) In term of considering stability of a method, it is the scaled stability region and not the stability region that is significant (cf., e.g., [1, p 198]) The stability region of an EPTRKN method is at the same time the scaled stability region With this stability interval, the associated EPTRKN method is expected to be efficient for solving problem (1.1) especially with a stringent accuracy demand The embedded EPTRKN pair 8(7) defined by (4.1) is implemented using local extrapolation and a starting procedure based on corrections until convergence of eight-stage direct collocation RKN method based on c8 (cf [21]) This embedded pair 8(7) gives an estimate of local truncation error of order given by LTE = y n+1 − y n+1 2 + y n+1 − y n+1 2 = O h8n , (4.2) where · denotes the Euclidean norm The stepsize strategy is similar to the one implemented by Sommeijer [25] in PIRKN codes which is also implemented in DOPRI5, DOP853 by Hairer and Wanner [20] A step is accepted when LTE TOL and rejected otherwise The new stepsize hn+1 is chosen as hn+1 = hn · 2, max 0.5, 0.8 · (TOL/LTE)1/8 (4.3) N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 141 The constants and 0.5 serve to keep the stepsize ratios hn+1 / hn to be in the interval [0.5, 2] This new EPTRKN method of order will be denoted by EPTRKN8 Furthermore, in the tables of numerical results, NSFCN and NPFCN denote the number of f -evaluations in sequential and parallel implementation modes, NCD is the number of correct decimal digits, NSTEP and NREJCT are the total number of integration steps and of rejected ones, respectively Our implementation of EPTRKN8 and ODEX2 has been carried out on a number of test examples They all lead to similar observations and therefore we report here only on the results of two examples The results of DOPRIN and PIRKN codes are reproduced from [25] 4.1 Nonlinear Fehlberg problem For the first numerical test, we apply the various codes ODEX2, DOPRIN, PIRKN and EPTRKN8 method to the well-known nonlinear Fehlberg problem (cf., e.g., [12,13,15,16]) d2 y(t) = dt −4t − 2 π/2 = (0, 1)T , y(t), y12 (t) + y22 (t) −4t y12 (t) + y22 (t) y T π/2 = −2 π/2, , y π/2 t 10, (4.4) with highly oscillating exact solution given by y(t) = (cos(t ), sin(t ))T The numerical results for this problem are listed in Table We see from this table that in parallel implementation mode, the EPTRKN8 method is the most efficient With stringent error tolerances EPTRKN8 method is even superior to ODEX2 and DOPRIN in a sequential computer 4.2 Newton’s equation of motion problem The second numerical example is the two-body gravitational problem for Newton’s equation of motion (see [24, p 245]) d2 y1 (t) =− dt y1 (t) y12 (t) + y22 (t) , d2 y2 (t) =− dt y2 (t) y12 (t) + y22 (t) , (4.5) y1 (0) = − ε, y2 (0) = 0, y1 (0) = 0, y2 (0) = 1+ε , 1−ε t 20 This problem can also be found in [16] or from the test set of problems in [22] The solution components are y1 (t) = cos u(t) − ε, y2 (t) = (1 + ε)(1 − ε) sin u(t) , where u(t) is the solution of Keppler’s equation t = u(t) − ε sin(u(t)) and ε denotes the eccentricity of the orbit In this example, we set ε = 0.9 The results reported in Table show a similar efficiency of the 142 N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 Table Numerical results for problem (4.4) Methods TOL NSFCN NPFCN NCD NSTEP NREJCT ODEX2 10−4 746 746 2.7 49 10−6 1122 1122 4.8 53 10−8 1493 1493 6.5 43 10−10 2039 2039 8.8 57 10−12 2907 2907 10.9 81 10−4 1288 161 1.2 154 26 10−6 1152 144 2.2 138 13 10−8 1168 146 6.7 140 10−10 2040 255 10.0 250 10−12 2736 342 10.5 337 10−4 633 633 3.8 79 10−8 2825 2825 8.3 353 51 10−12 9665 9665 12.3 1208 43 10−4 1800 300 3.9 50 10−8 3528 588 7.9 98 10−12 7422 1242 12.0 207 EPTRKN8 DOPRIN PIRKN EPTRKN8 method as for the Fehlberg problem when it is compared with ODEX2, DOPRIN and PIRKN codes Concluding remarks In this paper we have considered variable stepsize explicit pseudo two-step RKN methods requiring only one effective sequential f -evaluation per step for any order of accuracy Implemented with a variable stepsize strategy using embedding techniques, an explicit pseudo two-step RK methods derived from this class is shown to be superior to the currently most efficient sequential and parallel codes as ODEX2, DOPRIN and PIRKN In a stringent accuracy range, these methods are expected to have an efficiency equal if not superior to sequential codes even in a sequential implementation These conclusions encourage us to pursue the study of explicit pseudo two-step RKN methods In particular, we will concentrate on the optimal choice of the method parameters, numerical experiments with higherorder explicit pseudo two-step RKN methods and also on an implementation of these methods on parallel computers N.H Cong / Applied Numerical Mathematics 38 (2001) 135–144 143 Table Numerical results for problem (4.5) Methods TOL NSFCN NPFCN NCD NSTEP NREJCT ODEX2 10−4 691 691 1.4 62 15 10−6 1141 1141 3.6 85 18 10−8 1754 1754 5.9 95 20 10−10 2401 2401 7.4 101 20 10−12 3117 3117 9.7 115 18 10−4 1152 144 1.1 94 21 10−6 1472 184 2.5 152 45 10−8 2032 254 6.5 236 66 10−10 2552 319 10.0 311 33 10−12 3928 491 10.8 485 10−4 733 733 2.3 92 35 10−8 1545 1545 5.7 193 56 10−12 3777 3777 9.7 472 70 10−4 1836 306 1.2 51 19 10−8 2772 462 4.7 77 23 10−12 4716 786 8.9 131 33 EPTRKN8 DOPRIN PIRKN Acknowledgements A part of this work was done while I was a guest at the Institute of Numerical Mathematics, Halle University, Germany I would like to thank Prof Dr K Strehmel and Prof Dr R Weiner for their kind regards and interest in my research works I also would like to thank the referee for his/her useful comments which led to a further discussion concerning error estimates References [1] K Burrage, Parallel and Sequential Methods for Ordinary Differential Equations, Clarendon Press, 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Numerical Mathematics 38 (2001) 135–144 general class of explicit pseudo two-step RKN methods (EPTRKN methods) for solving problems of the form (1.1) A general s-stage (constant stepsize) EPTRKN method... from the literature Variable stepsize EPTRKN methods It is well known that an efficient integration method must be able to change stepsizes Because EPTRKN methods are of a two-step nature, there