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Applied Mathematics and Computation 217 (2011) 8010–8019 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Twostep-by-twostep continuous PIRKN-type PC methods for nonstiff IVPs q Nguyen Huu Cong ⇑, La Tri Dung Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam a r t i c l e i n f o a b s t r a c t In this paper, we start with the consideration of direct collocation-based Runge–KuttaNyström (RKN) methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y00 (t) = f(t, y(t)) At nth step, the continuous output formulas can be used for calculating the step values at (n + 2)th step and the integration processes can be proceeded twostep-by-twostep In this case, we obtain twostep-by-twostep RKN methods with continuous output formulas (continuous TBTRKN methods) Furthermore, we consider a parallel predictor–corrector (PC) iteration scheme using the continuous TBTRKN methods as corrector methods with predictor methods defined by the continuous output formulas The resulting twostep-bytwostep parallel-iterated RKN-type PC methods with continuous output formulas (twostep-by-twostep continuous PIRKN-type PC methods or TBTCPIRKN methods) give us a faster integration processes Numerical comparisons based on the solution of a few widely-used test problems show that the new TBTCPIRKN methods are much more efficient than the well-known PIRKN methods, the famous nonstiff sequential ODEX2, DOP853 codes and comparable with the CPIRKN methods Ó 2011 Elsevier Inc All rights reserved Keywords: Runge–Kutta–Nyström methods Predictor–corrector methods Stability Parallelism Introduction We consider numerical methods for solving nonstiff initial-value problems (IVPs) for the systems of special second-order, ordinary differential equations (ODEs) of the following form y00 tị ẳ ft; ytịị; yt0 ị ẳ y0 ; y0 t ị ẳ y0 ; t t T; d ð1:1Þ where y; f R Among various numerical methods proposed so far, the most efficient methods for solving the problems (1.1) are the explicit Runge–Kutta–Nyström (RKN) methods In the literature, sequential explicit RKN methods up to order 10 can be found in e.g., [19–23,25,26] With the arrival of parallel computers, several class of parallel predictor–corrector (PC) methods based on RKN corrector methods have been investigated in e.g., [3–11,15,16,31,12–14] A common challenge in the above-mentioned papers is to reduce, for a given order of accuracy, the required number of sequential f-evaluations per step, using parallel processors In the present paper, for constructing a particular class of parallel predictor–corrector (PC) methods, we consider RKN corrector methods with continuous output formulas From nth step, we can use the continuous output formulas for computing the step value at (n + 2)th step and apply a twostep-by-twostep integration strategy (the integration is proceeded twostep-by-twostep) In this way we obtain twostep-by-twostep RKN corrector methods with continuous output formulas (continuous TBTRKN corrector methods) Next we consider a parallel predictor–corrector (PC) iteration scheme using the continuous TBTRKN methods as correctors with predictors defined by the continuous output q This work was supported by the NAFOSTED Project 101.02.65.09 ⇑ Corresponding author Current address: School of Graduate Studies, VNU, 144 Xuan Thuy, Cau Giay, Hanoi, Viet Nam E-mail address: congnh@vnu.edu.vn (N.H Cong) 0096-3003/$ - see front matter Ó 2011 Elsevier Inc All rights reserved doi:10.1016/j.amc.2011.02.105 N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 8010–8019 8011 formulas The resulting parallel PC methods will be termed twostep-by-twostep parallel-iterated RKN-type PC methods with continuous output formulas (twostep-by-twostep continuous PIRKN-type PC methods or TBTCPIRKN methods) Thus, we have constructed parallel PC methods with continuous output formulas, high accuracy predictors and fast integration process Consequently, the resulting new TBTCPIRKN methods require few total numbers of sequential f-evaluations for a given accuracy In Section 2, we shall consider continuous TBTRKN corrector methods In Section 3, we formulate and investigate the TBTCPIRKN methods The investigations in Section concern the order of accuracy, the rate of convergence and the stability property Furthermore, in Section 4, we present numerical comparisons of TBTCPIRKN methods with parallel PIRKN and CPIRKN methods and sequential nonstiff ODEX2 and DOP853 codes Continuous TBTRKN corrector methods In the literature, efficient numerical methods are often equipped with a continuous output formulas which are especially useful in the case the number of output points becomes very large (cf [27, p 188]) Therefore, in the literature, efficient numerical methods are often provided with continuous output formulas In order to equip TBTCPIRKN methods considered in Section with a continuous output formula, we consider continuous TBTRKN corrector methods We begin with the consideration of a continuous extension of an implicit s-stage direct collocation (discrete) RKN method of the form (see e.g., [4,12,28]) Yn;i ẳ un ỵ hci u0n ỵ h s X aij ftn ỵ cj h; Yn;j ị; i ẳ 1; ; s; 2:1aị jẳ1 unỵ1 ẳ un ỵ hu0n þ h s X bj fðt n þ cj h; Yn;j ị; 2:1bị jẳ1 u0nỵ1 ẳ u0n ỵ h s X dj ft n ỵ cj h; Yn;j ị; 2:1cị jẳ1 and consider continuous output formulas dened by unỵn ẳ un ỵ nhu0n ỵ h s X bj nịft n ỵ cj h; Yn;j ị; 2:1dị jẳ1 u0nỵn ẳ u0n ỵ h s X dj nịft n ỵ cj h; Yn;j ị: 2:1eị jẳ1 Here, in (2.1), n 3; unỵn % yt nỵn ị; u0nỵn % y0 tnỵn ị, with tnỵn ẳ tn ỵ nh; unỵ1 % ytnỵ1 ị; un % ytn ị; u0nỵ1 % y0 tnỵ1 ị; u0n % y0 tn ị and h is the stepsize Furthermore, Yn,i, i = 1, , s are the stage vector components representing numerical approximations to the exact solution values y(tn + cih), i = 1, , s at nth step The s  s matrix A = (aij), s-dimensional vectors b = (bj), b(n) = (bj(n)), d = (dj), d(n) = (dj(n)) and c = (cj) are the method parameters in matrix or vector form The method defined by (2.1) will be called continuous RKN method (cf [13]) The step point and stage order of the (discrete) RKN method defined by (2.1a), (2.1b), (2.1c) will be referred to as the step point order p and the stage order r of the continuous RKN method By the collocation principle, the continuous RKN method (2.1) is of step point order p and stage order r both at least equal s (see [28]) If the (global) orders of continuous approximations defined by (2.1d) and (2.1e) are p0 and p1 respectively, then p⁄ = min{p0, p1} will be called the continuous order of the continuous RKN method (2.1) This continuous RKN method (2.1) can be conveniently presented by the Butcher tableau (see e.g., [2,13,28]) The matrix A and the vectors b, d, b(n) and d(n) are defined by the order conditions (see e.g., [12,28]) They can be explicitly expressed in terms of the collocation vector c as (see e.g., [12,13,17]) A ¼ PRÀ1 ; T b ¼ gT RÀ1 ; T ^T SÀ1 ; d ¼g ð2:2Þ 8012 N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 80108019 where ! cjỵ1 i ; jỵ1 P ẳ pij ị ẳ g ẳ g i Þ ¼    jÀ1 ; R ¼ ðrij ị ẳ jci  ; iỵ1 ^ ẳ g^i Þ ¼ g   ; i   S ẳ sij ị ẳ cj1 ; i i; j ¼ 1; ; s: The vectors b(n) and d(n) in the continuous output formula (2.1d) and (2.1e) respectively, are vector functions of n They have to satisfy the continuity conditions b(0) = 0, b(1) = b and d(0) = 0, d(1) = d and will be determined by order conditions which give (see also [13,17,18]) T b nị ẳ gT diagfn2 ; n3 ; ; nsỵ1 gR1 ; 2:3ị T ^T diagfn; n2 ; ; ns gS1 : d nị ẳ g The relations (2.2) and (2.3) imply that the continuity conditions for the vectors b(n) and d(n) are clearly verified For the step point order, stage order and continuous order of the continuous RKN method (2.1), we have the following theorem: Theorem 2.1 If the function f is Lipschitz continuous, then the step point order p, the stage order r and the continuous order p⁄ of the continuous RKN method (2.1) verify the following relations: p P s, r ¼ minfp; s ỵ 1g; p ẳ minfp; s ỵ 1g Proof The two first relations are implied from the order results for direct collocation-based RKN methods (cf [28]) The third relation can be obtained by applying the results in [17, Theorem 2.1] and in [13, Theorem 2.1] h Applying the approach used in [18], we now consider the following method Yn;i ẳ un ỵ ci hu0n ỵ h s X aij ftn ỵ cj h; Yn;j ị; i ẳ 1; ; s; 2:4aị jẳ1 unỵ2 ẳ un ỵ 2hu0n ỵ h s X bj 2ịftn ỵ cj h; Yn;j ị; 2:4bị jẳ1 u0nỵ2 ẳ u0n þ h s X dj ð2Þfðt n þ cj h; Yn;j ị: 2:4cị jẳ1 unỵn ẳ un ỵ nhu0n ỵ h s X bj nịft n ỵ cj h; Yn;j ị; 2:4dị jẳ1 u0nỵn ẳ u0n ỵ h s X dj nịft n ỵ cj h; Yn;j ị: 2:4eị j¼1 Here, in (2.4), bj(2) and dj(2), j = 1, , s are the components of the weight vectors b(2) and d(2) The formulas (2.4b) and (2.4c) compute (n + 2)th step values from nth step values Similar to the twostep-by-twostep implicit RK methods considered in [18], we shall call the method defined by (2.4a)–(2.4c) twostep-by-twostep implicit RKN method The method (2.4) will be referred to as continuous twostep-by-twostep RKN corrector method (continuous TBTRKN corrector method) and can also be conveniently presented by the Butcher tableau Definition 2.1 Suppose that un = y(tn) and u0n ¼ y0 ðtn Þ, then the continuous TBTRKN corrector method (2.4) is said to have ^ if ytnỵ2 ị unỵ2 ẳ Ohp^0 ỵ1 ị; y0 tnỵ2 ị u0nỵ2 ẳ Ohp^1 ỵ1 ị and p ^ ẳ minfp^0 ; p^1 g the step point order p We see that the integration process in the continuous TBTRKN corrector method (2.4) is proceeded twostep-by-twostep For the step point order, stage order and continuous order (orders of continuous approximations defined by (2.4d) and (2.4e)) of the continuous TBTRKN corrector method (2.4), we have the following theorem that is absolutely similar to Theorem 2.2 in [18] N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 8010–8019 8013 ^ ¼ s, the stage order ^r ¼ s and the Theorem 2.2 The continuous TBTRKN corrector method (2.4) has the step point order p ^à ¼ s continuous order p ^ ¼ s and ^r ¼ s are ensured by the collocation principle The third statement p ^à ¼ s can be Proof The two first statements p proved in the same way as in [17, Proof of Theorem 2.1] and in [13, Proof of Theorem 2.1] h TBTCPIRKN methods Let us consider in this section, a parallel PC iteration scheme based on the continuous TBTRKN corrector method (2.4) with the predictor method defined by the continuous output formulas This parallel PC iteration scheme is given by 0ị Yn;i ẳ yn2 ỵ þ ci Þhy0nÀ2 þ h s X ðmÞ bj ð2 þ ci Þfðt nÀ2 þ cj h; YnÀ2;j Þ; i ¼ 1; ; s; ð3:1aÞ j¼1 ðkÞ Yn;i ẳ yn ỵ ci hy0n ỵ h s X k1ị aij ft n ỵ cj h; Yn;j ị; i ¼ 1; ; s; k ¼ 1; ; m; 3:1bị jẳ1 ynỵ2 ẳ yn ỵ 2hy0n ỵ h   mị bj 2ịf tn ỵ cj h; Yn;j ; s X 3:1cị jẳ1 y0nỵ2 ẳ y0n ỵ h s X   mị dj 2ịf t n ỵ cj h; Yn;j ; 3:1dị jẳ1 ynỵn ẳ yn ỵ nhy0n ỵ h s X   mị bj nịf t n ỵ cj h; Yn;j ; 3:1eị jẳ1 y0nỵn ẳ y0n ỵ h s X   ðmÞ dj ðnÞf t n ỵ cj h; Yn;j ; 3:1fị jẳ1 where m is any number of iterations Regarding (3.1a) as predictor method and (2.4) as corrector method, we arrive at a PC   ðmÞ method in PE(CE)mE mode Since the evaluations of f t n2 ỵ cj h; Yn2;j ; j ẳ 1; ; s are available from the preceding twostep, we have in fact, a PC method in P(CE)mE mode In the PC method (3.1), the predictions given by (3.1a) are obtained by using the continuous output formula (3.1e) from the previous twostep Analogous to the continuous PIRKN-type PC methods (CPIRKN methods) considered in [13], we call the PC method (3.1) twostep-by-twostep CPIRKN method (TBTCPIRKN method)   k1ị ; j ẳ 1; ; s can be computed in parallel, provided that s processors We remark that the s components f t n ỵ cj h; Yn;j are available, so that the number of sequential f-evaluations per step of length h in each processor equals s⁄ = m + Theorem 3.1 If the function f is Lipschitz continuous, then for any number of iterations m, the TBTCPIRKN method (3.1) has step point order q = s and continuous order q⁄ = s Proof The proof of this theorem is very similar to the proofs of Theorem 3.1 in [13] and Theorem 3.1 in [17] Thus let us sỵ1 0ị suppose that f is Lipschitz continuous, yn = un = y(tn) and y0n ¼ u0n ¼ y0 ðtn Þ Since Yn;i À Yn;i ¼ Oðh Þ (cf Theorem 2.2) and each iteration raises the order of the iteration error by 2, we obtain the following (local) order relations mị Yn;i Yn;i ẳ Oh unỵ2 ynỵ2 ẳ h 2mỵsỵ1 s X i ẳ 1; ; s; ị; h i 2mỵsỵ3 mị bj 2ị ftn ỵ cj h; Yn;j ị ft n ỵ cj h; Yn;j ị ẳ Oh ị; jẳ1 u0nỵ2 y0nỵ2 ẳ h s X 3:2ị h i 2mỵsỵ2 mị dj 2ị ftn ỵ cj h; Yn;j ị ft n ỵ cj h; Yn;j ị ẳ Oh ị: jẳ1 Hence, for the local truncation error of the TBTCPIRKN method (3.1), we may write sỵ1 2mỵsỵ3 ytnỵ2 ị ynỵ2 ẳ ẵyt nỵ2 ị unỵ2 ỵ ẵunỵ2 ynỵ2 ẳ Oh ị ỵ Oh ị; sỵ1 2mỵsỵ2 0 0 ị: y t nỵ2 ị ynỵ2 ẳ y t nỵ2 ị unỵ2 ỵ unỵ2 ynỵ2 ẳ Oh ị ỵ Oh 3:3ị 8014 N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 8010–8019 The order relations (3.3) shows that the TBTCPIRKN method (3.1) has the step point order q = s for any number of iterations m as stated in Theorem 3.1 Furthermore, for the continuous order q⁄ of the TBTCPIRKN method, we have yt nỵn ị ynỵn ẳ ẵyt nỵn ị unỵn ỵ ẵunỵn ynỵn ẳ ẵyt nỵn ị unỵn ỵ ẵun yn ỵ hn u0n y0n s h i X mị ỵh bj nị ft n ỵ cj h; Yn;j ị ft n ỵ cj h; Yn;j ị ; jẳ1 y0 t nỵn ị y0nỵn ẳ y0 t nỵn ị u0nỵn þ u0nþn À y0nþn  à  à ¼ y t nỵn ị u0nỵn ỵ u0n y0n s h i X mị dj nị ft n ỵ cj h; Yn;j ị ft n ỵ cj h; Yn;j ị : ỵh 3:4ị jẳ1 From Theorem 2.2 and the order relations (3.2), we obtain the following global order relations yt nỵn ị unỵn ẳ Oh sỵ1 ị; sỵ1 y0 t nỵn ị u0nỵn ẳ Oh ị; s h i X 2mỵsỵ3 mị bj nị ft n ỵ cj h; Yn;j ị ftn ỵ cj h; Yn;j ị ẳ Oh ị; h 3:5ị jẳ1 h s X h i 2mỵsỵ2 mị dj nị ft n þ cj h; Yn;j Þ À fðtn þ cj h; Yn;j ị ẳ Oh ị: jẳ1 The relations (3.4) and (3.5) then complete the Proof of Theorem 3.1 h Remark By virtue of Theorem 3.1, the various orders of the TBTCPIRKN methods will not increase if the number of iterations m increases So that in (3.1), by setting m = 0, we obtain the cheapest PC methods which require only one sequential f-evaluation per step However, in practice, the TBTCPIRKN methods are often implemented with nonzero m in order to achieve an acceptable stability and to compensate for the iteration errors 3.1 Rate of convergence As for all explicit parallel RKN-type PC methods, the rate of convergence of TBTCPIRKN methods is also defined by using the model test equation y00 (t) = ky(t), where k runs through the eigenvalues of the Jacobian matrix @f/@y (cf., e.g., [4,6,7,13,14]) Applying (3.1b) to the model test equation, we obtain the iteration error equation h i ðjÀ1Þ YðjÞ À Yn ; n À Yn ¼ zA Y n z :¼ h k; j ¼ 1; ; m: ð3:6Þ Hence, with respect to the model test equation, the convergence rate is determined by the spectral radius q(zA) of the iteration matrix zA Requiring that q(zA) < 1, leads us to the convergence condition jzj < qðAÞ or h < qð@f=@yÞqðAÞ : ð3:7Þ We shall call q(A) the convergence factor and 1/q(A) the convergence boundary of the TBTCPIRKN methods One can exploit the freedom in the choice of the collocation vector c of continuous TBTRKN corrector methods for minimizing the convergence factor q(A), or equivalently, for maximizing the convergence region denoted by Sconv and defined as Sconv :¼ fz : jzj < 1=qðAÞg: ð3:8Þ The convergence factors q(A) for the TBTCPIRKN methods which will be used in the numerical comparisons can be found in Section 3.2 Stability intervals The linear stability of the TBTCPIRKN methods (3.1) is investigated by again using the model test equation y00 (t) = ky(t), where k is assumed to be lying in the negative real axis Let us define the matrix B ¼ ðbð2 þ c1 Þ; ; bð2 þ cs ÞÞT :  T ð0Þ ð0Þ By using the matrix B, the starting stage vector Yð0Þ defined by (3.1a) for the model test equation, can be n ¼ Y n;1 ; ; Y1;s presented in the form 8015 N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 80108019 mị Y0ị n ẳ eyn2 ỵ 2e ỵ cịhyn2 ỵ zBYn2 ; z :ẳ h k: Using this formula for Yð0Þ n and applying (3.1b)–(3.1d) to the model test equation yield À 0Á YðmÞ ẳ eyn ỵ chyn ỵ zAYm1ị ẳ ẵI ỵ zA þ Á Á Á þ ðzAÞmÀ1 Š eyn þ chyn ỵ zAịm Y0ị n n n 0 ẳ zmỵ1 Am BYn2 ỵ ẵI ỵ zA ỵ ỵ zAịm1 eyn ỵ ẵI ỵ zA ỵ ỵ zAịm1 chyn ỵ zm Am eyn2 ỵ zm Am 2e ỵ cịhyn2 mị 3:9aị T ynỵ2 ẳ yn ỵ 2hyn ỵ zb 2ịYmị n T T T ẳ zmỵ2 b 2ịAm BYn2 ỵ f1 ỵ zb 2ịẵI ỵ zA ỵ ỵ zAịm1 egyn ỵ f2 ỵ zb 2ịẵI ỵ zA ỵ ỵ zAịm1 cghyn mị T T ỵ zmỵ1 b 2ịAm eyn2 ỵ zmỵ1 b 2ịAm 2e ỵ cịhyn2 ; 3:9bị T hynỵ2 ẳ hyn ỵ zd 2ịYnmị T T T ẳ zmỵ2 d 2ịAm BYn2 ỵ zd 2ịẵI ỵ zA ỵ ỵ zAịm1 eyn ỵ f1 ỵ zd 2ịẵI ỵ zA ỵ ỵ zAịm1 cghyn mị T T þ zmþ1 d ð2ÞAm eynÀ2 þ zmþ1 d ð2ÞAm ð2e þ cÞhynÀ2 : ð3:9cÞ Relations (3.9) lead us to the recursion YðmÞ n ðmÞ YnÀ2 C C B B B ynỵ2 C B yn C C B C B B hy C ẳ Mm zịB hy0 C; B nỵ2 C B n C C C B B @ yn A @ ynÀ2 A 0 hyn hynÀ2 ð3:10aÞ where Mm(z) is an (s + 4)  (s + 4) matrix defined by PmÀ1 ðzÞe Pm1 zịc z m Am e zm Am 2e ỵ cị zmỵ1 Am B C B mỵ2 T B z b 2ịAm B ỵ zbT 2ịPm1 zịe ỵ zbT 2ịPm1 zịc zmỵ1 bT 2ịAm e zmỵ1 bT 2ịAm 2e ỵ cị C C B C B T T T T M m zị ẳ B zmỵ2 dT 2ịAm B zd 2ịPm1 zịe ỵ zd 2ịPm1 zịc zmỵ1 d 2ịAm e zmỵ1 d 2ịAm 2e ỵ cị C: C B C B 0 0T A @ 0T 0 ð3:10bÞ Here in (3.10b), PmÀ1(z) = I + zA + Á Á Á + (zA)mÀ1 The matrix Mm(z) defined by (3.10b) which determines the stability of the TBTCPIRKN methods, will be called the amplification matrix, its spectral radius q(Mm(z)) the stability function For a given number of iterations m, the stability interval denoted by (Àbstab(m), 0) of the TBTCPIRKN methods is defined as bstab mị; 0ị :ẳ fz : qM m zịị < 1; z 0g: We shall call bstab(m) the stability boundary for a given m The stability boundaries bstab(m) for the TBTCPIRKN methods which will be used in the numerical comparisons can be found in Section 4 Numerical comparisons This section will investigate numerical comparison for the TBTCPIRKN methods We consider the TBTCPIRKN methods based on s-stage direct collocation-based continuous TBTRKN corrector methods with the collocation vector c = (c1, , cs)T whose the components ci are the roots of the first-kind Chebyshev polynomial of degree s in the interval [0, 2] i.e ci ¼ cos   2i p ỵ 1; 2s i ¼ 1; ; s: For the collocation-based continuous TBTRKN methods, this choice of collocation points seems to be reasonable We not claim that this choice of collocation points is the best In a forthcoming paper, we will investigate variable stepsize TBTCPIRKN methods with the optimal choice of collocation points for collocation-based continuous TBTRKN corrector methods In this paper, we confine our considerations to the fixed stepsize TBTCPIRKN methods denoted by TBTCPIRKN4 and TBTCPIRKN6 respectively based on the continuous TBTRKN corrector methods of and stages The step point order, stage order and continuous order of the two resulting TBTCPIRKN methods are all equal to and 6, respectively (cf Theorem 3.1) The convergence factors q(A) as defined in Section 3.1 of TBTCPIRKN4 and TBTCPIRKN6 methods are computed to be respectively equal to 0.117 and 0.061 The stability boundaries of these TBTCPIRKN methods are listed in Table Notice that the stability boundaries of the two TBTCPIRKN methods show a rather regular behavior when compared with other parallel ex- 8016 N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 8010–8019 plicit RKN-type methods (cf e.g., [4–8,10,11,31,12–14]) These stability boundaries are not really large, however they are sufficiently large for nonstiff IVPs of the form (1.1) In the following, we shall compare the above TBTCPIRKN methods with explicit parallel RKN-type methods and sequential codes from the literature For the TBTCPIRKN methods, in the first step, we always use the trivial predictions given by ð0Þ Y0;i ẳ y0 ỵ hci y00 ; i ẳ 1; ; s: The absolute error obtained at the end point of the integration interval is presented in the form 10ÀNCD (NCD may be interpreted as the average number of correct decimal digits) The computational efforts are measured by the values of NFUN denoting the total number of sequential f-evaluations required over the total number of integration steps Ignoring load balancing factors and communication times between processors in parallel methods, in the numerical comparisons, a method is considered more efficient if for a given computational cost defined by NFUN, it can give higher accuracy defined by NCD or equivalently, for a given accuracy defined by NCD, it requires fewer computational cost defined by NFUN The numerical experiments with small widely-used test problems taken from the literature below show a potential superiority of the new TBTCPIRKN methods over existing methods This superiority will be significant in a parallel machine if the test problems are large enough and/or the f-evaluations are expensive (cf., e.g., [1]) In order to see the convergence behavior of TBTCPIRKN methods in the PC iteration process (3.1b), we follow a dynamical strategy for determining the number of iterations in the successive steps It seems natural to require that the iteration error is of the same order in h as the order of the corrector methods This leads us to the stopping criterion (cf., e.g., [3,4,6–8,10]) p ðmÀ1Þ kYmị k1 TOL ẳ Ch ; n Yn ð4:1Þ where C is a problem- and method-dependent parameter, p is the step point order of the corrector method All the computations were carried out on a 15-digit precision computer 4.1 Problems used in the numerical comparison We use a set of three well-known test problems taken from the ODE literature These three problems possess exact solutions in closed form Initial conditions are taken from the exact solutions LINE – The linear nonautonomous problem (cf e.g., [4,6,7]) d2 yðtÞ dt  ẳ  2atị ỵ atị ỵ ytị; 2atị 1ị atị 4:2ị atị ẳ maxf2cos2 tị; sin2 tịg; y0ị ẳ 0; 0ịT ; y0 0ị ẳ 1; 2ịT ; t 20; T with exact solution y(t) = (Àsin(t), 2sin(t)) FEHL – The0nonlinear Fehlberg problem (cf e.g., [19,20,22,23]) d2 yðtÞ dt B ¼@ À4t 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 y1 tịỵy2 tị p y p=2ị ẳ 0; 1ịT ; p 2 y1 tịỵy2 tị C Aytị; 4t2 p p y0 p=2ị ẳ p=2; 0ÞT ð4:3Þ pffiffiffiffiffiffiffiffiffi p=2 t 10; with highly oscillating exact solution given by y(t) = (cos(t2), sin(t2))T NEWT – The two-body gravitational problem for Newton’s equation of motion (see e.g., [30, p 245],[23,29]) d y1 ðtÞ dt y1 tị ẳ q 3 ; y1 tị ỵ y22 tị y1 0ị ẳ e; y2 0ị ẳ 0; d y2 tị d t y01 0ị ẳ 0; y2 tị ẳ q 3 ; y1 tị ỵ y22 tị r 1ỵe ; t 20: y2 0ị ¼ 1Àe ð4:4Þ Table Stability boundaries bstab(m) for various TBTCPIRKN methods Methods TBTCPIRKN4 TBTCPIRKN6 bstab(1) bstab(2) bstab(3) bstab(4) bstab(5) bstab(6) 0.800 0.643 2.210 2.315 2.280 2.288 0.664 0.513 2.374 2.387 2.385 2.385 N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 8010–8019 8017 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The solution components are y1(t) = cos(u(t)) e, y2 tị ẳ þ eÞð1 À eÞsinðuðtÞÞ, where u(t) is the solution of Kepler’s equation t = u(t) À esin(u(t)) and e denotes the eccentricity of the orbit In order to make this problem nonstiff, we set e = 0.3 4.2 Comparison with parallel methods We confine on the comparison of the TBTCPIRKN methods considered in this paper with ones of the best parallel explicit RKN methods available in the literature, that is the PIRKN methods proposed in [4,31] and the CPIRKN methods investigated in [13] The abscissas used in CPIRKN methods are obtained by halving the abscissas used in TBTCPIRKN methods We consider indirect PIRKN (INPIRKN) methods investigated in [31] and direct PIRKN (DPIRKN) methods investigated in [4] As for TBTCPIRKN methods, we restrict our consideration to the INPIRKN, DPIRKN and CPIRKN methods of order and order respectively denoted by INPIRKN4, DPIRKN4, CPIRKN4, INPIRKN6, DPIRKN6 and CPIRKN6 TBTCPIRKN4, TBTCPIRKN6 and all these INPIRKN, DPIRKN, CPIRKN methods are implemented with the fixed stepsize and with the same stopping criterion (4.1) The INPIRKN, DPIRKN, CPIRKN and TBTCPIRKN methods are applied to the above three test problems The values of NFUN are plotted as a function of the values of NCD The results in Figs 1–3 show that TBTCPIRKN4 method is competitive with CPIRKN4 method and more efficient than INPIRKN4 and DPIRKN4 methods These results also show that TBTCPIRKN6 method is competitive with CPIRKN6 method and superior to INPIRKN6 and DPIRKN6 methods 4.3 Comparison with sequential codes In Section 4.2, 4th-order TBTCPIRKN4 method and 6th-order TBTCPIRKN6 method were compared with 4th-order and 6th-order indirect, direct PIRKN and CPIRKN methods In this section, we shall compare the TBTCPIRKN methods with some Fig Results for LINE Fig Results for FEHL 8018 N.H Cong, L.T Dung / Applied Mathematics and Computation 217 (2011) 8010–8019 Fig Results for NEWT Fig Comparison with DOP853 and ODEX2 of the best sequential codes currently available in the literature We restricted our consideration to the numerical comparison of our TBTCPIRKN6 method with two well-known sequential nonstiff codes for FEHL (the nonlinear Fehlberg problem (4.3)), that is the codes DOP853 and ODEX2 taken from [27] We apply TBTCPIRKN6 and ODEX2 directly to FEHL, DOP853 to the first order form of FEHL The obtained values of NCD and NFUN are plotted in Fig In spite of the fact that the results of DOP853 and ODEX2 codes are obtained using a stepsize strategy, whereas TBTCPIRKN6 method is applied with fixed stepsizes, it is the TBTCPIRKN6 method that is the most efficient (see Fig 4) Concluding remarks In this paper, we investigated a new class of parallel PC methods called twostep-by-twostep continuous parallel-iterated RKN-type PC methods (TBTCPIRKN methods) The numerical comparisons based on the solution of three often-used test problems taken from the ODE literature showed that the TBTCPIRKN methods are competitive with the CPIRKN methods and by far superior to the well-known indirect, direct PIRKN methods and the famous codes DOP853 and ODEX2 The paper limits its focus to problems of the form (1.1), however, there has been proposed RKN methods for the more general problem y00 ðxÞ ¼ fðt; 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Applied Mathematics and Computation 217 (2011) 8010–8019 8011 formulas The resulting parallel PC methods will be termed twostep-by-twostep parallel-iterated RKN-type PC methods with continuous. .. with continuous output formulas (twostep-by-twostep continuous PIRKN-type PC methods or TBTCPIRKN methods) Thus, we have constructed parallel PC methods with continuous output formulas, high accuracy... Parallel-iterated RK-type PC methods with continuous output formulas, Int J Comput Math 80 (2003) 1027–1037 N.H Cong, L.N Xuan, Twostep-by-twostep PIRK-type PC methods with continuous output formulas,