DSpace at VNU: Improved parallel-iterated pseudo two-step RK methods for nonstiff IVPs

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Applied Numerical Mathematics 58 (2008) 160–170 www.elsevier.com/locate/apnum Improved parallel-iterated pseudo two-step RK methods for nonstiff IVPs ✩ Nguyen Huu Cong a,∗ , Le Ngoc Xuan b a School of Graduate Studies, Vietnam National University, Hanoi G7, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam b Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Available online 16 January 2007 Abstract The aim of this paper is to consider a parallel predictor–corrector (PC) iteration scheme for a general class of pseudo two-step Runge–Kutta methods (PTRK methods) of arbitrary high-order for solving first-order nonstiff initial-value problems (IVPs) on parallel computers Starting with an s-stage pseudo two-step RK method of order p∗ with w implicit stages, we apply a highly parallel PC iteration process in PE(CE)m E mode The resulting parallel PC method can be viewed as a parallel-iterated pseudo twostep Runge–Kutta method (PIPTRK method) with an improved (new) predictor formula and therefore will be called the improved PIPTRK method (IPIPTRK method) The IPIPTRK method uses an optimal number of processors equal to w p∗ /2 Numerical experiments show that the IPIPTRK methods proposed in this paper are superior to the efficient sequential DOPRI5 and DOP853 codes and parallel PIRK methods available in the literature © 2006 IMACS Published by Elsevier B.V All rights reserved MSC: 65M12; 65M20 Keywords: RK methods; PC methods; Parallelism Introduction The arrival of parallel computers influences the development of numerical methods for the solution of the nonstiff initial value problem (IVP) for systems of first-order ordinary differential equations (ODEs) y (t) = f t, y(t) , y(t0 ) = y0 , t0 t T, (1.1) where y, f ∈ Rd The most efficient numerical methods for solving this problem are the explicit Runge–Kutta methods (RK methods) In the literature, sequential explicit RK methods up to order 10 can be found in e.g., [16,18,19] In order to exploit the facilities of parallel computers, a number of parallel explicit methods have been investigated in e.g., [1–4,6–9,11–15,20–22] 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 the present paper, we propose a class of parallel predictor–corrector (PC) iteration methods based on the class of pseudo two-step ✩ This work was supported by NRPFS * Corresponding author E-mail address: congnh@vnu.edu.vn (N.H Cong) 0168-9274/$30.00 © 2006 IMACS Published by Elsevier B.V All rights reserved doi:10.1016/j.apnum.2006.11.005 N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 161 RK methods (PTRK methods) recently proposed in [10] An s-stage PTRK method using w implicit stages, v = s − w explicit stages is of step point and stage order both at least equal to s with any integer pair w, v, with w + v = s It is always zero-stable and can attain the step point order p ∗ = s + (see Section 2) Applying a highly parallel PC iteration scheme to the PTRK methods gives us the parallel PC methods that are similar to the parallel-iterated pseudo two-step RK methods (PIPTRK methods) proposed in [12] with an improved (new) predictor formula Therefore, the resulting PC iteration methods will be termed improved parallel-iterated PTRK methods (IPIPTRK methods) (see Section 3) Although, for a given number of processors, the order of the IPIPTRK methods used for the numerical experiments in this paper is equal to that of the PIRK methods, their rate of convergence is better, their predictor formula is more accurate, so that their efficiency is expected to be increased (see Section 4) The increased efficiency is demonstrated in Sections 4.1 and 4.2 by comparing numerical results of the IPIPTRK methods with those of PIRK and sequential explicit RK methods available in the literature In the following sections, for the sake of simplicity of notation, we assume that the IVP (1.1) is a scalar problem However, all considerations below can be straightforwardly extended to a system of ODEs PTRK corrector methods The PTRK corrector methods were firstly considered in [10] In this section, we give an overview on these PTRK methods Let collocation vector c be partitioned into two subvectors cv and cw that is c = (cTv , cTw )T , then a general s-stage PTRK method based on c for a scalar problem is defined by Vn = un ev + hAvv f (tn−1 ev + hcv , Vn−1 ) + hAvw f (tn−1 ew + hcw , Wn−1 ), (2.1a) Wn = un ew + hAwv f (tn ev + hcv , Vn ) + hAww f (tn ew + hcw , Wn ), (2.1b) un+1 = un + hbTv f (tn ev (2.1c) + hcv , Vn ) + hbTw f (tn ew + hcw , Wn ), where, un+1 ≈ y(tn+1 ), Aij are i × j method parameter matrices, bj , dj and cj are j -dimensional method parameter vectors, ej is the j -dimensional vector with unit entries, (for i, j = v, w, v + w = s) Vn is called the explicit stage subvector representing the numerical approximation to the exact solution vector y(tn ev + cv h) = [y(tn + c1 h), , y(tn + cv h)]T and Wn is called the implicit stage subvector representing the numerical approximation to the exact solution vector y(tn ew + cw h) = [y(tn + cv+1 h), , y(tn + cs h)]T Furthermore, in (2.1) and elsewhere in this paper, we use for any two vectors ξ = (ξ1 , , ξs )T , η = (η1 , , ηs )T and any scalar function f the notation f (ξ , η) := [f (ξ1 , η1 ), , f (ξs , ηs )]T The method parameter matrices Aij and vectors bj (for i, j = v, w, v + w = s), will be determined by order conditions (see Theorem 2.1 below) This PTRK method is conveniently specified by the tableau Avv Owv Avw Oww cv cw un+1 Ovv Awv bTv Ovw Aww bTw (2.2) In order to start the PTRK method (2.1), an appropriate starting procedure is needed for generating sufficiently accurate starting stage vectors V0 , W0 and step point value u1 from u0 = y0 This can be done, for example, by using an appropriate PIRK method considered in [21] or a sequential RK code with dense output The s-stage PTRK method (2.1) consists of v explicit stages and w implicit stages Its order can be studied in the same way as the order of RK methods Thus suppose that un = y(tn ), Vn−1 = y(tn−1 ev + hcv ) and Wn−1 = y(tn−1 ew + hcw ), then we have the following order definition (see [10]): Definition 2.1 The PTRK method (2.1) is said to be of order p ∗ if y(tn+1 ) − un+1 = O hp and stage order q∗ ∗ +1 = min{p ∗ , q , , q2 } if in addition, y(tn ev + hcv ) − Vn = O hq1 +1 , y(tn ew + hcw ) − Wn = O hq2 +1 162 N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 The following theorem gives the order conditions for PTRK methods (see [10, Theorem 2.1]) Theorem 2.1 If the function f is Lipschitz continuous, and if (Avv , Avw )(c − e)j −1 = (Awv , Aww )cj −1 = j cv , j j = 1, , q1 , (2.3a) j cw , j bTv , bTw cj −1 = , j j = 1, , q2 , (2.3b) j = 1, , p, (2.3c) then the PTRK method (2.1) has order p ∗ = min{p, q1 + 1, q2 + 1} and stage order q ∗ = min{p ∗ , q1 , q2 } for any collocation vector c with distinct abscissas and for any integer pair v, w with w + v = s In order to express the parameter matrices Avv , Avw , Awv , Aww and vectors bv , bw explicitly in terms of the collocation vector c = (cTv , cTw )T , we define the following matrices and vectors Pv := cv c2v c3v c4v cs , , , , , v , s gT := R := e, c, c2 , c3 , , cs−1 , cw c2w c3w c4w cs , , , , , w , s 1 , , , , s Pw := Q := e, (c − e), (c − e)2 , , (c − e)s−1 Then the order conditions (2.3) in Theorem 2.1 for q1 = q2 = p = s can be presented in the form (cf [10]) (Avv , Avw )Q = Pv , bTv , bTw R = gT (Awv , Aww )R = Pw , (2.4) From (2.4) the parameter matrices and vectors of the PTRK method (2.1) can be expressed as follows (Avv , Avw ) = Pv Q−1 , (Awv , Aww ) = Pw R −1 , bTv , bTw = gT R −1 (2.5) In view of Theorem 2.1, it follows from (2.5) that y(tn ev + hcv ) − Vn = O hs+1 , y(tn ew + hcw ) − Wn = O hs+1 , y(tn + h) − un+1 = O hp+1 (2.6) By virtue of Theorem 2.1, p = s for any collocation vector c with distinct abscissas With a special choice of the vector c, it is possible to increase the order p beyond s (superconvergence) by satisfying an orthogonality relation (see e.g., [19, p 212]) as stated in the following theorem: Theorem 2.2 An s-stage PTRK method defined by (2.1) is of step point order p ∗ = s and of stage order q ∗ = s if the parameter matrices Avv , Avw , Awv , Aww and vectors bv , bw , of the method satisfy the relations (2.5) (equivalently (2.4)) for any collocation vector c with distinct abscissas and for any integer pair v, w with w + v = s It has step point order p ∗ = s + if in addition x Pj (1) = 0, ξ j −1 Pj (x) := holds for k s (ξ − ci ) dξ, j = 1, , k, i=1 The proof of this theorem can be found in [10] Theorem 2.2 indicates that an s-stage PTRK method, can attain the step point order p ∗ = s + According to the analysis of the local errors in this section, the starting vectors V0 , W0 and value u1 should be of order s + and p ∗ + that is N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 163 y(t0 ev + hcv ) − V0 = O hs+1 , y(t0 ew + hcw ) − W0 = O hs+1 , y(t1 ) − u1 = O hp ∗ +1 Since the PTRK method (2.1) is of two-step nature, the property of zero-stability is an important requirement One can ask whether PTRK methods are zero-stable The following theorem gives the answer to this question Theorem 2.3 The PTRK methods based on any collocation vectors c with distinct abscissas are always zero-stable for any integer pair v, w with v + w = s This theorem was proved in [10] by writing the PTRK methods in the one-step form of a general linear method (see e.g., [5]) IPIPTRK methods In this section, by applying a parallel PC iteration scheme to the PTRK methods defined by (2.1), we arrive at the following PC iteration method (in PE(CE)m E mode): (m) Vn = yn ev + hAvv f (tn−1 ev + hcv , Vn−1 ) + hAvw f tn−1 ew + hcw , Wn−1 , W(0) n = yn ew (j ) Wn = yn ew + hBwv f (tn ev + hcv , Vn ) + hBww f + hAwv f (tn ev + hcv , Vn ) + hAww f yn+1 = yn + hbTv f (tn ev + hcv , Vn ) + hbTw f (m) tn−1 ew + hcw , Wn−1 , (j −1) , tn ew + hcw , Wn tn ew + hcw , W(m) n (3.1a) (3.1b) j = 1, , m, (3.1c) (3.1d) , where the matrices Bvw and Bww in (3.1b) are determined by order conditions given in Section 3.1 This PC iteration method (3.1) is similar to a PIPTRK method considered in [12] and becomes the original PIPTRK method if the “improved” predictor formula (3.1b) in (3.1) is replaced with the original one used in [12]: (m) W(0) n = yn ew + hBwv f (tn−1 ev + hcv , Vn−1 ) + hBww f tn−1 ew + hcw , Wn−1 The PC method (3.1) therefore, will be termed the improved PIPTRK method (IPIPTRK method) As for every explicit method, the computational cost of the method (3.1) is measured by the number of sequential f-evaluations per step Notice that v components of f (tn ev + hcv , Vn ) and w components of f (tn ew + j −1 hcw , Wn ), j = 1, , m + 1, can be computed in parallel, provided that max(v, w) processors are available Since f (tn−1 ev + hcv , Vn−1 ) and f (tn ew + hcw , Wn−1 ) can be reused, in general, we need m + sequential f-evaluations per step 3.1 Order conditions for the predictor (0) The sth order conditions for the predictor formula (3.1b) can be derived by replacing Vn , yn , Wn−1 and Wn by the exact solution values y(tn ev + hcv ), y(tn ), y(tn−1 ew + hcw ) = y(tn ew + h(cw − ew )), and y(tn ew + hcw ), respectively On substitution of these exact solution values into (3.1b), we are led to y(tn ew + hcw ) − y(tn )ew − hBwv y (tn ev + hcv ) − hBww y tn ew + h(cw − ew ) = O hs+1 (3.2) Using a Taylor expansion for the sufficiently smooth function y(t) in the neighborhood of tn , we can expand the left-hand side of (3.2) in powers of h and obtain the order conditions (cf., e.g., [8,10,12]) cjw − j Bwv cvj −1 + Bww (cw − ew )j −1 = 0, j = 1, , s, (3.3a) or equivalently Bwv cvj −1 + Bww (cw − ew )j −1 = j cw , j j = 1, , s (3.3b) 164 N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 The conditions (3.3) determine the matrix (Bwv , Bww ) By using the matrices Pw defined in Section and the new matrix ev cv c2v cs−1 v Q1 = , ew (cw − ew ) (cw − ew )2 (cw − ew )s−1 these conditions (3.3) can be written in the form (Bwv , Bww )Q1 = Pw (3.4) Suppose that the following condition is satisfied: ci = cj − 1, for i = 1, , v and j = v + 1, , s (3.5a) Since the condition (3.5a) implies that the matrix Q1 is nonsingular, the relation (3.4) leads us to the explicit expression of the matrix (Bwv , Bww ) as follows (Bwv , Bww ) = Pw Q−1 (3.5b) If (3.4) (equivalently (3.3)) is satisfied, then we have (0) = O hs+1 Wn − W(0) n = Wn − y(tn ew + hcw ) + y(tn ew + hcw ) − Wn (3.6) Since the function f is Lipschitz continuous and each iteration in (3.1) raises the order of the iteration error by 1, using (3.6) we can obtain the following order relations: m+s+1 , Wn − W(m) n =O h un+1 − yn+1 = hbTw f (Wn ) − f W(m) n = O hm+s+2 , y(tn+1 ) − yn+1 = y(tn+1 ) − un+1 + [un+1 − yn+1 ] = O hp ∗ +1 + O hm+s+2 , where, p ∗ is the order of the PTRK corrector method (2.1) Since condition (3.5) implies the relations (3.3) (equivalently (3.4)), we have the following theorem: Theorem 3.1 If the PTRK corrector method (2.1) has step point order p ∗ , and if the conditions (3.5) are satisfied, then the IPIPTRK method (3.1) has step point order p ∗∗ = min{p ∗ , m + s + 1}, for any collocation vector c with distinct abscissas 3.2 Rate of convergence The rate of convergence of the IPIPTRK method (3.1) is defined by using the model test equation y (t) = λy(t), where λ runs through the eigenvalues of the Jacobian matrix ∂f/∂y (cf., e.g., [7,11,14,20]) Applying the IPIPTRK method (3.1) to this model test equation, we obtain the iteration error equation (j −1) (j ) Wn − Wn = zAww Wn − Wn , z := hλ, j = 1, , m (3.7) Hence, with respect to the model test equation, the rate of convergence is determined by the spectral radius ρ(zAww ) of the iteration matrix zAww Requiring that ρ(zAww ) < 1, leads us to the convergence condition |z| < ρ(Aww ) or h< ρ(∂f/∂y)ρ(Aww ) (3.8) We shall call ρ(Aww ) the convergence factor and 1/ρ(Aww ) the convergence boundary of the IPIPTRK methods The freedom in the choice of the collocation vector c of PTRK corrector methods can be used for minimizing the convergence factor ρ(Aww ), or equivalently, for maximizing the convergence region Sconv defined by Sconv := z: |z| < 1/ρ(Aww ) (3.9) Specification of convergence factors and convergence boundaries for a specified class of IPIPTRK methods used in our numerical experiments is reported in Section N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 165 3.3 Stability regions The linear stability of the IPIPTRK methods (3.1) is investigated by again using the model test equation y (t) = λy(t), where λ ∈ C− := {z: z ∈ C, Re(z) 0} Denoting z := λh and applying (3.1) to the model test equation yields (m) Vn = zAvv Vn−1 + zAvw Wn−1 + ev yn , (3.10a) m−1 (ew yn + zAwv Vn ) + (zAww )m W(0) W(m) n = I + zAww + · · · + (zAww ) n (m) = I + zAww + · · · + (zAww )m−1 (ew yn + zAwv Vn ) + (zAww )m ew yn + zBwv Vn + zBww Wn−1 (m) = I + zAww + · · · + (zAww )m−1 ew yn + zAwv ev yn + zAvv Vn−1 + zAvw Wn−1 (m) (m) + (zAww )m ew yn + zBwv ev yn + zAvv Vn−1 + zAvw Wn−1 + zBww Wn−1 = z2 I + zAww + · · · + (zAww )m−1 Awv Avv + zm+2 (Aww )m Bwv Avv Vn−1 (m) + z2 I + zAww + · · · + (zAww )m−1 Awv Avw + zm+1 (Aww )m zBwv Avw + Bww Wn−1 + I + zAww + · · · + (zAww )m−1 (ew + zAwv ev ) + zm (Aww )m [ew + zBwv ev ] yn (m) (m) (m) (m) = M21 (z)Vn−1 + M22 (z)Wn−1 + M23 (z)yn , (3.10b) yn+1 = yn + zbTv Vn + zbTw W(m) n (m) (m) (m) (m) (m) = yn + zbTv zAvv Vn−1 + zAvw Wn−1 + ev yn + zbTw M21 (z)Vn−1 + M22 (z)Wn−1 + M23 (z)yn (m) (m) (m) (m) = z2 bTv Avv + zbTw M21 (z) Vn−1 + z2 bTv Avw + zbTw M22 (z) Wn−1 + + zbTv ev + zbTw M23 (z) yn (m) (m) (m) (m) = M31 (z)Vn−1 + M32 (z)Wn−1 + M33 (z)yn From (3.10) we are led to the recursion ⎞ ⎛ ⎞ ⎛ Vn−1 Vn ⎜ (m) ⎟ ⎝ W(m) n ⎠ = Mm (z) ⎝ Wn−1 ⎠ , yn+1 (3.10c) (3.11a) yn where Mm (z) is the (s + 1) × (s + 1) matrix defined by ⎞ ⎛ zA zAvw ev vv (m) (m) ⎟ ⎜ (m) Mm (z) = ⎝ M21 (z) M22 (z) M23 (z) ⎠ (m) M31 (z) (3.11b) (m) (m) M32 (z) M33 (z) (m) of Mij (z) with i = 2, 3, The explicit formulas j = 1, 2, in (3.11b) are clear from (3.10) The matrix Mm (z) in (3.11) which determines the stability of the IPIPTRK methods, will be called the amplification matrix, its spectral radius ρ(Mm (z)) the stability function For a given number of iterations m, the stability region of the IPIPTRK methods are defined as Sstab (m) := z: ρ Mm (z) < 1, Re(z) The real and imaginary boundaries for a given m, βre (m) and βim (m), respectively, can be defined in the familiar way The stability pairs (βre (m), βim (m)) for the IPIPTRK methods used in our numerical experiments can be found in Section 4 Numerical experiments In this paper, we report numerical results for IPIPTRK methods with the number of implicit stages w = [s/2] and the number of explicit stages v = s − w, where [·] denotes the integer part function and s = 4, 5, (see Section 3) The PTRK corrector methods defined by (2.1) are based on collocation vectors c which satisfy the condition (3.5a) We restrict our numerical experiments to the three following IPIPTRK methods: 166 N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 • The first method denoted by IPIPTRK4 uses the PTRK corrector based on collocation vector c = (cTv , cTw )T with √ √ cv = (c1 , c2 )T , cw = ( 65 + c1 , 65 + c2 )T , where c1 = (3 − )/6 and c2 = (3 + )/6 are two components of the two-dimensional Gauss–Legendre collocation vector The resulting IPIPTRK4 method is of order p ∗∗ = (see Theorem 2.2 and Theorem 3.1) and uses pr = w = processors • The second method denoted by IPIPTRK5 uses the PTRK corrector vector c = (cTv , cTw )T √ based on collocation √ 6 T T with cv = (c1 , c2 , c3 ) , cw = ( + c1 , + c2 ) , where c1 = (4 − )/10, c2 = (4 + )/10 and c3 = are three components of the three-dimensional RadauIIA collocation vector The resulting IPIPTRK5 method is of order p ∗∗ = (see Theorems 2.2 and 3.1) and uses pr = w = processors T T )T with • The third method denoted by IPIPTRK6 uses the PTRK corrector based w √ on collocation vector c = (cv , c√ 6 T T cv = (c1 , c2 , c3 ) , cw = ( + c1 , + c2 , + c3 ) , where c1 = (5 − 15 )/10, c2 = 1/2 and c3 = (5 + 15 )/10 are three components of the three-dimensional Gauss–Legendre collocation vector The resulting IPIPTRK6 method is of order p ∗∗ = (see Theorems 2.2 and 3.1) and uses pr = w = processors We not claim that the above chosen IPIPTRK methods are the best possible for a given order p ∗∗ A further study of this topic will be subject of future research The orders and number of processors pr of IPIPTRK4 and IPIPTRK6 methods are the same as used for the PIRK methods proposed in [21] with pr = p ∗∗ /2 except for the IPIPTRK5 method with pr < p ∗∗ /2 A direct numerical computation reveals that the convergence factors of the IPIPTRK methods as defined in Section 3.2 are slightly smaller than those of PIRK methods of the same order (see Table 1, where * indicates that no PIRK method of order p = is available) Table also gives the convergence boundaries defined by 1/ρ(Aww ) (see Section 3.2) of the IPIPTRK methods Table lists the real and imaginary stability boundaries of the IPIPTRK methods We note that all these IPIPTRK methods have no empty imaginary stability boundaries The results reported in Tables and show that for the IPIPTRK methods, the convergence boundaries are larger than stability boundaries βre (m) and βim (m) for p = 4, 5, and m = 1, , Hence, the benefit of large stability regions can be fully utilized As shown in the Table 2, the stability pairs of the IPIPTRK methods are sufficiently large for nonstiff problems We shall compare the IPIPTRK methods with parallel and sequential explicit RK methods from the literature In the numerical experiments, for the first step, the starting values V0 , W0 and y1 of an IPIPTRK method will be generated by the associated PIRK method based on the same collocation vector c as the underlying IPIPTRK method The absolute error obtained at the end point of the integration interval is presented in the form 10−NCD (NCD may be interpreted as the number of correct decimal digits) The computational efforts are measured by the values of Nseq denoting the total number of sequential f-evaluations required over the total number of integration steps Nstp Ignoring load balancing factors and communication times between processors in parallel methods, the comparison of various methods in this section is based on the scaled accuracy NCD/Nseq The numerical experiments with small widely-used test problems taken from the literature below show a potential superiority of the new IPIPTRK methods Table Convergence factors and boundaries for various pth order parallel PC methods pth order parallel PC methods p=4 p=5 p=6 Convergence factors for PIRK (cf [7]) Convergence factors for IPIPTRK Convergence boundaries for IPIPTRK 0.289 0.275 3.636 * 0.169 5.917 0.215 0.204 4.901 Table Stability pairs (βre (m), βim (m)) for various pth order IPIPTRK methods pth order IPIPTRK methods p=4 p=5 p=6 m=1 m=2 m=3 m=4 m=5 m=6 (0.483, 0.068) (1.033, 0.069) (1.659, 0.069) (1.795, 0.069) (1.883, 0.069) (1.965, 0.069) (0.285, 0.294) (0.766, 0.794) (1.255, 1.375) (1.871, 1.907) (2.282, 2.444) (3.241, 2.485) (0.197, 0.199) (0.609, 0.203) (1.070, 0.203) (1.456, 0.203) (1.738, 0.203) (1.942, 0.203) N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 167 over extant 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., [3]) All the computations were carried out on a 15-digit precision computer An actual implementation with stepsize strategy for large and expensive problems on a parallel machine is a subject of further studies 4.1 Comparison with parallel PC methods In order to see the efficiency of various parallel PC methods, we follow a dynamical strategy in all PC methods 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 underlying corrector methods This leads us to the stopping criterion (cf., e.g., [7,11]) (m−1) W(m) n − Wn ∗ ∞ TOL = Chp , (4.1) p∗ where C is a problem- and method-dependent parameter, and is order of the PTRK correctors We shall report numerical results obtained by ones of the best parallel explicit RK methods available in the literature, that is the PIRK methods proposed in [21] and the three IPIPTRK methods constructed in this paper We selected a test set of three problems taken from the literature 4.1.1 Two body problem As a first numerical test, we apply the various pth order parallel PC methods to the two body problem on the (cf., e.g.,[21,23]) integration interval [0, 20], with eccentricity ε = 10 y1 (t) = y3 (t), y1 (0) = − ε, y2 (0) = 0, y2 (t) = y4 (t), −y4 (t) , y3 (t) = [y1 (t) + y22 (t)]3/2 y4 (t) = −y2 (t) , [y1 (t) + y22 (t)]3/2 y3 (0) = 0, y4 (0) = 1+ε 1−ε (4.2) The numerical results listed in Table clearly show that the IPIPTRK methods are much more efficient than the PIRK methods of the same order and the same number of processors For two-processor parallel PC methods considered here, the IPIPTRK5 method is the best 4.1.2 Fehlberg problem For the second numerical test, we apply the various pth order parallel PC methods to the often-used Fehlberg problem on the integration interval [0, 5] (cf.,e.g., [7,21,23]) y1 (t) = 2ty1 (t) log max y2 (t), 10−3 , y2 (t) = −2ty2 (t) log max y1 (t), 10 −3 y1 (0) = 1, , y2 (0) = e, (4.3) with the exact solution y1 (t) = exp(sin(t )), y2 (t) = exp(cos(t )) The numerical results are reported in Table These numerical results show that the IPIPTRK methods are again by far superior to the PIRK methods of the same order and the same number of processors The IPIPTRK5 method is again the best of two-processor parallel PC methods Table Values of NCD/Nseq for problem (4.2) obtained by various pth order parallel PC methods with pr processors PC methods p pr Nstp = 100 Nstp = 200 Nstp = 400 Nstp = 800 Nstp = 1600 PIRK IPIPTRK4 4 2 3.1/441 2.4/212 3.7/905 3.7/405 4.9/1947 5.0/804 6.1/4000 6.2/1604 7.3/8000 7.4/3204 IPIPTRK5 4.4/206 5.6/405 7.0/805 8.5/1605 10.0/3205 PIRK IPIPTRK6 6 3 5.0/643 5.8/264 7.2/1302 7.4/479 8.9/2637 9.1/920 10.5/5499 10.8/1840 12.3/11200 12.4/3480 168 N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 Table Values of NCD/Nseq for problem (4.3) obtained by various pth order parallel PC methods with pr processors PC methods p pr Nstp = 100 Nstp = 200 Nstp = 400 Nstp = 800 Nstp = 1600 PIRK IPIPTRK4 4 2 2.7/392 3.5/213 4.0/842 4.8/412 5.2/1756 6.0/803 6.5/3650 7.2/1601 7.7/7409 8.4/3201 IPIPTRK5 4.2/227 5.8/431 7.3/833 8.8/1620 10.3/3202 PIRK IPIPTRK6 6 3 5.2/601 6.2/274 7.0/1245 8.4/503 8.9/2542 10.3/942 10.7/5199 12.6/1825 12.5/10488 Table Values of NCD/Nseq for problem (4.4) obtained by various pth order parallel PC methods with pr processors PC methods p pr Nstp = 100 Nstp = 200 Nstp = 400 Nstp = 800 PIRK IPIPTRK4 4 2 2.3/300 4.4/202 5.1/800 6.3/403 6.3/1600 7.5/803 7.5/3200 8.8/1603 Nstp = 1600 8.9/6571 10.0/3203 IPIPTRK5 6.1/203 8.5/403 9.6/804 11.1/1604 12.7/3204 PIRK IPIPTRK6 6 3 5.1/486 8.7/206 7.8/1126 10.2/405 11.2/2345 11.9/805 12.5/4775 13.7/1605 4.1.3 Jacobian elliptic functions problem The final numerical example is the Jacobian elliptic functions sn, cn, dn problem for the equation of motion of a rigid body without external forces on the integration interval [0, 20] (cf., e.g., [19, Problem JACB, p 240], also [23]) y1 (t) = y2 (t)y3 (t), y1 (0) = 0, y2 (t) = −y1 (t)y3 (t), y3 (t) = −0.51y1 (t)y2 (t), y2 (0) = 1, y3 (0) = (4.4) The exact solution is given by the Jacobi elliptic functions y1 (t) = sn(t; k), y2 (t) = cn(t; k), y3 (t) = dn(t; k) (see [17]) The numerical results for this problem are given in Table and give rise to nearly the same conclusions as formulated in the two previous examples 4.2 Comparison with sequential methods In Section 4.1, the IPIPTRK methods were compared with PIRK methods In this section, we shall compare these IPIPTRK methods with some of the best sequential explicit RK methods currently available In order to compare the methods of comparable order, we restricted the numerical experiments to the comparison of our 5th and 6th order IPIPTRK5 and IPIPTRK6 methods with two sequential codes DOPRI5 and DOP853 of 5th and 8th order, respectively, for the Fehlberg problem (4.3) These DOPRI5 and DOP853 codes are embedded explicit RK methods due to Dormand and Prince and coded by Hairer and Waner (see [19]) They are based on the pair 5(4) and the “triple” 8(5)(3), respectively DOP853 is the new version of DOPRI8 with a “stretched” error estimator (see [19, p 254]) These two codes belong to the most efficient currently existing sequential codes for nonstiff first-order ODE problems We took the best results obtained by DOPRI5 and DOP853 given in [9] and added the results in the low accuracy range obtained by IPIPTRK5 and IPIPTRK6 methods In spite of the fact that the results of the sequential codes are obtained using a stepsize strategy, whereas IPIPTRK5 and IPIPTRK6 methods are applied with fixed stepsizes, it is the IPIPTRK5 and IPIPTRK6 methods are more efficient when compared to the sequential codes of comparable order (see Table 6) From Table 6, we can see that the IPIPTRK5 method is about two and half times cheaper than the code DOPRI5, and the IPIPTRK6 method is about two times cheaper than the code DOP853 N.H Cong, L.N Xuan / Applied Numerical Mathematics 58 (2008) 160–170 169 Table Comparison with sequential methods for problem (4.3) Methods Nstp NCD Nseq DOPRI5 (from [19]) 75 162 393 979 2458 3.2 5.3 7.4 9.4 11.4 452 974 2360 5876 14 750 DOP853 (from [19]) 47 70 107 164 261 4.5 6.2 8.0 10.2 12.2 552 825 1265 1950 3123 IPIPTRK5 (in this paper) 100 200 400 800 1600 3200 4.2 5.8 7.3 8.8 10.3 11.9 227 431 833 1620 3202 6402 IPIPTRK6 (in this paper) 50 100 200 400 800 3.6 6.2 8.4 10.3 12.6 162 274 503 942 1825 Concluding remarks In this paper, we proposed and investigated a new class of parallel PC iteration methods called improved paralleliterated pseudo two-step RK methods (IPIPTRK methods) that are shown to be promising parallel integration methods Three numerical experiments clearly demonstrate the superiority of the IPIPTRK methods over the efficient sequential DOPRI5 and DOP853 codes and parallel PIRK methods available in the literature In forthcoming papers, we will pursue the study of IPIPTRK methods via the optimal choice of the method parameters and variable stepsize control References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] K Burrage, 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parallel-iterated pseudo two-step RK methods (PIPTRK methods) proposed in [12] with an improved (new) predictor formula... Concluding remarks In this paper, we proposed and investigated a new class of parallel PC iteration methods called improved paralleliterated pseudo two-step RK methods (IPIPTRK methods) that are shown... iteration of pseudo two-step RK methods for nonstiff IVPs, Japan J Indust Appl Math 20 (2003) 51–64 N.H Cong, H Podhaisky, R Weiner, Numerical experiments with some explicit pseudo two-step RK methods