DSpace at VNU: Parallel-iterated pseudo two-step Runge-Kutta methods with step size control

DSpace at VNU: Parallel-iterated pseudo two-step Runge-Kutta methods with step size control tài liệu, giáo án, bài giảng...

Japan J Indust Appl Math (2014) 31:441–460 DOI 10.1007/s13160-014-0144-6 ORIGINAL PAPER Area Parallel-iterated pseudo two-step Runge–Kutta methods with step size control Nguyen Huu Cong · Nguyen Thu Thuy Received: 14 November 2012 / Revised: 25 April 2014 / Published online: 14 May 2014 © The JJIAM Publishing Committee and Springer Japan 2014 Abstract The aim of this paper is to develop a class of constant step size paralleliterated pseudo two-step Runge–Kutta methods (PIPTRK methods) for nonstiff firstorder ODE problems into variable step size methods Embedded formulas are provided for giving a cheap error estimate used in the step size control Methods with variable parameters approach were applied for overcoming the difficulty in using two-step methods with variable step size By applications to a few widely used test problems, we compare the efficiency of the resulting PIPTRK methods with step size control (PIPTRKSC methods) with the codes PIRK, DOPRI5, DOP853 and ODEX This numerical comparison shows that these new PIPTRKSC methods are by far superior to the PIRK, DOPRI5, DOP853 and ODEX codes Keywords Runge–Kutta methods · Two-step Runge–Kutta methods · Predictor–corrector methods · Parallelism Mathematics Subject Classification (2010) 65M12 · 65M20 This work was supported by the NAFOSTED Project 101.02-2013.06 N H Cong · N T Thuy Faculty of Mathematics, Mechanics and Informatics , Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan , Hanoi, Vietnam Present address: N H Cong (B) School of Graduate Studies , Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam e-mail: congnh@vnu.edu.vn 123 442 N H Cong, N T Thuy Introduction The advent of parallel computers influences the development of methods for the numerical solution of a nonstiff initial value problem (IVP) for a system 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., [18,20,22] In order to exploit the facilities of parallel computers, a number of parallel explicit methods have been investigated in e.g., [1–8,10–17,23–25] A common challenge in the latter mentioned works is to reduce, for a given order of accuracy, the required number of effective sequential fevaluations per step, using parallel processors In the previous paper [11], the authors have considered a general class of constant step size parallel-iterated pseudo twostep RK methods (constant step size PIPTRK methods) for solving nonstiff IVPs of the form (1.1) The constant step size PIPTRK methods are obtained by applying a parallel predictor–corrector (PC) iteration scheme to pseudo two-step RK methods (PTRK methods) investigated in [9] These constant step size PIPTRK methods were shown to be more efficient when compared with existing methods (see [11]) In the present paper, we equip the constant step size PIPTRK methods considered in [11] with an ability of being able to control the step sizes during the integration process We start our considerations in Sect 2, with embedded variable step size PTRK methods including the order conditions and two embedded formulas Section investigates PIPTRK methods with step size control (PIPTRKSC methods) based on the parallel PC iteration of the embedded variable step size PTRK methods Here in this section, the order conditions for the predictor and the rate of convergence are considered The better efficiency of PIPTRKSC methods is demonstrated in Sect by comparing numerical results of the PIPTRKSC methods with those of PIRK, DOPRI5, DOP853 and ODEX codes 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 Embedded variable step size PTRK methods The constant step size PTRK methods were firstly considered in [9] In this paper, we consider variable step size PTRK methods with embedded formulas For twostep nature methods, there is an additional difficulty in applying them to a variable step size mode There exist in principe two approaches for overcoming this difficulty (cf., e.g., [3, p 44], [22, p 397]): (i) interpolating past stage values, (ii) deriving methods with variable parameters The first approach using polynomial interpolation to reproduce the stage values for the new step, involves with computational cost, which increases as the dimension of the problems increases, while for the second 123 Parallel-iterated pseudo two-step RK methods 443 approach, the computational cost is independent of the dimension of the problems For this reason, the variable parameter approach is more feasible and robust Thus, we consider the variable step size PTRK methods with variable parameters Suppose that cv and cw have respective dimensions sv and sw with sv + sw = s For scalar problems, a general s-stage variable step size PTRK method based on collocation T )T has the following form vector c = (cvT , cw Vn = u n ev + h n Anvv f (tn−1 ev + h n−1 cv , Vn−1 ) + h n Anvw f (tn−1 ew + h n−1 cw , Wn−1 ), (2.1a) Wn = u n ew + h n Awv f (tn ev + h n cv , Vn ) + h n Aww f (tn ew + h n cw , Wn ), (2.1b) T u n+1 = u n + h n bvT f (tn ev + h n cv , Vn ) + h n bw f (tn ew + h n cw , Wn ), (2.1c) where, h n = tn+1 − tn , u n+1 ≈ y(tn+1 ), Anvv , Anvw , Awv and Aww respectively are (sv × sv )-dimensional, (sv × sw )-dimensional, (sw × sv )-dimensional and (sw × sw )dimensional matrices, bv and bw respectively are sv -dimensional and sw -dimensional vectors, ev and ew respectively are sv -dimensional and sw -dimensional vectors with unit entries Vn is called the explicit stage subvector representing the numerical approximation to the exact solution vector y(tn ev +cv h n ) = [y(tn +c1 h n ), , y(tn +csv h n )]T and Wn is called the implicit stage subvector representing the numerical approximation to the exact solution vector y(tn ew + cw h n ) = [y(tn + csv +1 h n ), , y(tn + cs h n )]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 Anvv , Anvw , Awv , Aww and vectors bv , bw will be determined by order conditions This variable step size PTRK method is conveniently specified by the tableau Anvv Owv Anvw Oww cv cw u n+1 Ovv Awv bvT Ovw Aww T bw In (2.1), the two-step formula (2.1a) has the variable parameter matrices Anvv and which depend on n In order to proceed from u to u using the variable step size PTRK method (2.1), an appropriate starting procedure is needed for generating sufficiently accurate the stage vectors V0 and W0 This can be done, for example, by using an appropriate PIRK method considered in [24] or a sequential RK code with dense output The s-stage variable step size PTRK method (2.1) consists of sv explicit stages and sw implicit stages Its step point order and stage order can be studied in the same way as the step point order and stage order of the constant step size PTRK methods investigated in [9] Thus suppose that u n = y(tn ), Vn−1 = y(tn−1 ev + h n−1 cv ) and Wn−1 = y(tn−1 ew + h n−1 cw ), then we have the following order definition (cf [9]): Anvw , Definition 2.1 The variable step size PTRK method (2.1) is said to be of order p ∗ if p ∗ +1 y(tn+1 ) − u n+1 =O(h n ), 123 444 N H Cong, N T Thuy and stage order q ∗ = min{ p ∗ , q1 , q2 } if in addition, q +1 y(tn ev + h n cv ) − Vn = O(h n1 q +1 ), y(tn ew + h n cw ) − Wn = O(h n2 ) 2.1 Order conditions The qth-order conditions for (2.1a), (2.1b) and pth-order conditions for (2.1c) can be derived by replacing Vn−1 , Wn−1 , u n , Vn , Wn and u n+1 in (2.1) with the exact solution values y(tn−1 ev +h n−1 cv ) = y(tn ev +h n−1 (cv −ev )), y(tn−1 ew +h n−1 cw ) = y(tn ew + h n−1 (cw −ew )), y(tn ), y(tn ev +h n cv ), y(tn ew +h n cw ) and y(tn+1 ), respectively, that is y(tn ev + h n cv ) − y(tn )ev − h n Anvv y (tn ev + h n−1 (cv − ev )) q+1 −h n Anvw y (tn ew + h n−1 (cw − ew )) = O(h n ), y(tn ew + h n cw ) − y(tn )ew − h n Awv y (tn ev + h n cv ) q+1 −h n Aww y (tn ew + h n cw ) = O(h n y(tn+1 ) − y(tn ) − h n bvT y (2.2a) ), T (tn ev + h n cv ) − h n bw y (2.2b) (tn ew + h n cw ) = p+1 O(h n ) (2.2c) Let us suppose that the step size ratio h n / h n−1 is bounded from above (i.e., ), then along the same lines of [9, Section 2.1], using Taylor expanh n / h n−1 sions, we can expand the left-hand side of (2.2) in powers of h n and obtain the order conditions for determining the method parameter matrices and vectors given by C(v j) = hn h n−1 j−1 j cv − (Anvv , Anvw )(c − e) j−1 = 0, j j = 1, , q, (2.3a) j C(wj) = cw − (Awv , Aww )c j−1 = 0, j D ( j) = T − (bvT , bw )c j−1 = 0, j j = 1, , q, j = 1, , p (2.3b) (2.3c) We have the following theorem (cf [9, Theorem 2.1]): Theorem 2.1 If the function f is Lipschitz continuous, and if the order conditions (2.3) are satisfied, then the s-stage variable step size PTRK method (2.1) has step point order p ∗ = min{ p, q + 1} and stage order q ∗ = min{ p, q} for any collocation T )T with distinct abscissas and for any integer pair s , s with vector c = (cvT , cw v w sv + sw = s Proof Suppose that f is Lipschitz continuous and u n = y(tn ), Vn−1 = y(tn−1 ev + , the principal error coefh n−1 cv ), Wn−1 = y(tn−1 ew + h n−1 cw ) For h n / h n−1 (q+1) (q+1) ( p+1) , Cw and error term D in (2.3) are uniformly bounded Conseficients Cv quently, similar to the order considerations for a general variable step size multistep method (cf., e.g., [22, p 401]), the conditions (2.3a) and (2.3b) imply that locally 123 460 N H Cong, N T Thuy 20 Hairer, E.: A Runge–Kutta method of order 10 J Inst Math Appl 21, 47–59 (1978) 21 Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations (I) In: Nonstiff Problems, 1st edn Springer-Verlag, Berlin (1987) 22 Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations (I) In: Nonstiff Problems, 2nd edn Springer-Verlag, Berlin (1993) 23 van der Houwen, P.J., Cong, N.H.: Parallel block predictor–corrector methods of Runge–Kutta type Appl Numer Math 13, 109–123 (1993) 24 van der Houwen, P.J., Sommeijer, B.P.: Parallel iteration of high-order Runge–Kutta methods with stepsize control J Comput Appl Math 29, 111–127 (1990) 25 van der Houwen, P.J., Sommeijer, B.P.: Block Runge–Kutta methods on parallel computers Z Angew Math Mech 68, 3–10 (1992) 26 Hull, T.E., Enright, W.H., Fellen, B.M., Sedgwick, A.E.: Comparing numerical methods for ordinary differential equations SIAM J Numer Anal 9, 603–637 (1972) 123 ... variable step size PTRK methods including the order conditions and two embedded formulas Section investigates PIPTRK methods with step size control (PIPTRKSC methods) based on the parallel PC iteration... the constant step size PIPTRK methods considered in [11] with an ability of being able to control the step sizes during the integration process We start our considerations in Sect 2, with embedded... iteration scheme to pseudo two -step RK methods (PTRK methods) investigated in [9] These constant step size PIPTRK methods were shown to be more efficient when compared with existing methods (see [11])