P.J. van der Houwen and B.P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control, J. Comput. Appl. Math. 29 (1990), 111-127.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... methods or explicit Runge Kutta methods (cf. e.g. Butcher [6] Hairer et al. 12] a number of parallel explicit methods (cf. e.g. Burrage [1, 2] Burrage and Suhartanto [5] Chu and Hamilton [7] Cong [8] Cong and Mitsui [11] van der Houwen and Cong [14] van der Houwen and Sommeijer [15, 16], Jackson and Nrsett [18] Lie [19] Nrsett and Simonsen [21] etc. has been proposed for exploiting new computing facilities. In a recent work of Cong [9] a general class of explicit pseudo two step RK methods (EPTRK methods) for solving problems of the form (1.1) has been considered. This ....

....pairs in numerical tests on a sequential computer. For the first step a starting procedure based on corrections until convergence of an appropriate s stage collocation RK corrector is used. The stepsize strategy in our codes is similar to the one implemented by van der Houwen and Sommeijer [15] in PIRK methods which is also implemented in PIMRK methods by Burrage and Suhartanto [5] and in DOPRI5, DOP853 by Hairer and Wanner [12] The new stepsize h n 1 is chosen as h n 1 = h n Delta min ae 3; max ae 0:3; 0:8 Delta kerrk Gamma1=p oeoe ; 3.3) where p is the local order of ....

P.J. van der Houwen and B.P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control, J. Comput. Appl. Math. 29 (1990), 111-127.

....IVP requires a solution method with good stability properties [7] Usually those methods include the solution of a large number of implicit systems which is very expensive. A class of solution methods called iterated Runge Kutta methods have been proposed for a parallel solution of IVPs [18] 8] [21] [24] Iterated Runge Kutta methods are predictor corrector (PC) methods based on implicit Runge Kutta (RK) correctors, i.e. the corrector steps represent an iteration of the (implicit) basic RK method. These methods have a large degree of inherent parallelism and are therefore very attractive ....

....solutions are provided which allow to control the stepsizes without further computational effort. The stability properties of iterated RK methods depend on the way the corrector is iterated. A functional iteration (fixed point iteration) of an implicit RK corrector results in the IRK method. In [21] and [23] IRK methods were proposed for a parallel implementation on shared memory machines with a small number s of processors (s is the number of stages of the corrector RK method) In [16] IRK methods has been parallelized for distributed memory machines. But because of their relatively ....

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P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111-- 127, 1990.

....1 Introduction Many applications in the area of scientific computing exhibit different levels of potential parallelism. Examples from the area of numerical analysis are solution methods for ordinary differential equations (ODEs) like extrapolation methods [24] iterated Runge Kutta (RK) methods [37, 26], or implicitly iterated RK methods [38, 23] These methods compute several independent approximation vectors in each time step, which can be computed in parallel to each other and which are combined to determine the final solution vector. Examples from physical simulation include ....

....if these rows are not stored on the same processor. The new iteration vector z (j 1) is computed by all processors. Thus, in the next Newton iteration step, the iteration vector z (j 1) is available on all processors required for the evaluation of F . 5. 2 Example 2: IRK method The IRK method [37] solves a system of ordinary differential equations y 0 = f(y) in a time interval [x 0 ; x end ] with a number of time steps. Each time step consists of a fixed number m of corrector steps. This results in the following iteration scheme: oe (0) l = j ; l = 1; s ; 7) oe (j) l = j ....

P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111--127, 1990.

....algorithms exhibit a two level structure of potential method and system parallelism. Examples in the area of numerical analysis methods are solution methods for ordinary differential equations (ODEs) like extrapolation methods, iterated Runge Kutta (RK) methods, or implicitly iterated RK methods [26, 27]. These methods compute several independent approximation vectors in each time step, which can be computed in parallel to each other and which are combined to determine the final solution vector. In contrast to this medium grain method parallelism, examples from physics often exhibit coarsegrain ....

....: IR Theta IR n R I n is an application specific nonlinear function. The predefined vector y 0 = y 0;1 ; y 0;n ) specifies the initial condition at the point x 0 . Iterated RK methods are derived from implicit RK methods by functional iteration of the implicit system to be solved [26]. Each macrostep of an s stage iterated RK method executes a fixed number m of corrector steps each consisting of the evaluation of s expressions containing the right hand side f of the ODE system. This results in the following iteration scheme for one macrostep: oe (0) l = j ; l = 1; ....

P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111--127, 1990.

....the prediction mechanism for the SP2. 5. 2 The Iterated Runge Kutta Method The iterated RK method is an explicit one step method for the solution of initial value problems which is derived from an s stage implicit RK method by iterating the stage vector system for a fixed number of times m [18]. The resulting approximations of the stage vectors are then used to compute the new iteration vector. When considering systems of differential equations a potential data parallelism can be exploited. The iterated RK method has an additional potential of parallelism due to the independence of the ....

P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111--127, 1990.

.... dt = Gamma l ff l X jjjn ff l Delta (OE nh j ; h l ) l = 0; Sigma1; Sigman (4) fi l = 1 l (j n j 2 ; h l ) l = Sigma1; Sigman (5) 3 Solving the ODE system The ODE system (4) is solved with a parallel diagonal implicitly iterated Runge Kutta (DIIRK) method [3, 5]. One time step of the DIIRK method consists of a predictor step and a fixed number m of corrector steps. An s stage, implicit Runge Kutta method, the base RK method, is used as corrector. v (0) l = y l = 1; s (6) v (j) l = y h s X i=1 (a li Gammad li )f (v (j Gamma1) i ....

....y h s X l=1 b l f (v (m) l ) 8) The s dimensional vectors b = b 1 ; b s ) and c = c 1 ; c s ) and the s Theta s matrix A = a li ) describe the base RK method. The convergence order of the DIIRK method is r = min(r; m 1) where r is the order of the base RK method [5]. For the computation of v (j) l we have to solve the nonlinear system F j;l = 0 with F j;l : IR n IR n defined as follows F j;l (z) z Gamma y Gamma h s X i=1 (a li Gammad li )f (v (j Gamma1) i ) Gamma hd ll f (z) l = 1; s; j = 1; m: 9) The m nonlinear implicit systems ....

P. van der Houwen and B. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. J. of Comp. and Appl. Math., 29:111--127, 1990. This article was processed using the L a T E X macro package with LLNCS style

....methods because they are much more expensive than linear multistep correctors. The advantage of using RK methods are smaller error constants. Furthermore, the PC iterations with implicit RK methods provide a high degree of parallelism. Van der Houwen and Sommeijer suggest IRK methods in [8] and [7] for a parallel execution on a shared memory machines. They concentrate on mathematical characteristics (stability, convergence order) of the methods and don t give a runtime analysis or predict or measure speedup values. In this article, we propose parallel versions for the IRK method with ....

....and the resulting runtimes for the IRK method. The last section 5 contains the comparisons of the runtimes and the results of the practical implementation on the iPSC=860. 2 Iterated Runge Kutta Methods We describe predictor corrector methods with s stage, implicit, one step RK correctors, [7], 8] y 1 = y h s X l=1 b l v l where y = y(x 0 H) The n dimensional vector v l , l = 1; s, are defined implicitly by the following system of equations of dimension s Delta n: v l = f(y h s X i=1 a li v i ) l = 1; s The s dimensional vector b = b ....

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P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111--127, 1990.

....(fixed point iteration) explicit Runge Kutta methods are obtained which are appropriate for the numerical solution of nonstiff systems (1) Embedded solutions are used for the stepsize control. Van der Houwen and Sommeijer suggest IRK methods for a parallel execution on a shared memory machines [6]. They concentrate on mathematical characteristics (stability, convergence order) of the methods and not on runtime analysis or prediction or measurement of speedup values. The investigation of implementations of IRK methods on distributed memory machines (DMMs) in [5] shows that the attainable ....

..... 3 The Iterated Runge Kutta Method In this section, we summarize the IRK method and its parallel implementation using the general notation of equation (1) 3. 1 Numerical Method The IRK method is a one step predictor corrector method that uses an s stage, implicit, one step RK corrector, [6]. An RK method is given by y 1 = y h s X l=1 b l v l where y = y(x 0 H) is an approximation of the value y(x 0 H) and v l , l = 1; s, are implicitly defined by the nonlinear system: v l = f(y h s X i=1 a li v i ) l = 1; s In case of a system of ODE (1) ....

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P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111--127, 1990.

....x Gamma1 h one after another with step size h. There is a large variety of RK methods that can be used for ODEs with different characteristics. Here, we consider RK methods with a large potential of task and data parallelism. These methods have been especially designed for a parallel execution [33, 22]. We apply the solution methods to two classes of ODEs which differ in the amount of computational work of the right hand side f of the ODE system: ffl f has fixed evaluation costs that are independent of the system size (sparse function) ffl the evaluation costs of f depend linearly on the ....

P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111-- 127, 1990.

.... to the methods considered in [4, 6] the PTRK methods also fall into the class of general linear methods (GLM) cf. e.g. 2, 11] However, the parameters of PTRK methods are easily to be derived and the methods themself are applicable to parallel computations based on the approach used in e.g. [3, 5, 7, 12, 13]. In Section 2 we will define and investigate the PTRK methods where order conditions, zero stability aspect and an option for the choice of the method parameters are considered. Furthermore, in Section 3, we present a numerical test showing the theoretical performance of PTRK methods. In the ....

P.J. van der Houwen and B.P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control, J. Comput. Appl. Math. 29 (1990), 111-127.

....in the k th block can be computed simultaneously by solving independent systems of m equations each. Consequently, the ERK and DIRK variants, respectively, retain their characteristic explicit or diagonally implicit property. Regrettably, these ERK schemes offer little potential for parallelism [21, 27], as the order of a p block ERK formula cannot exceed p and, if the order is p, the stability region is fz : j P p i=0 z i =i j 1g, which is not large. The construction of parallel ERK formulas and some minor advantages of these schemes are discussed in [21, 27] However, a ....

....potential for parallelism [21, 27] as the order of a p block ERK formula cannot exceed p and, if the order is p, the stability region is fz : j P p i=0 z i =i j 1g, which is not large. The construction of parallel ERK formulas and some minor advantages of these schemes are discussed in [21, 27]. However, a predictor corrector variant discussed below seems more promising. On the other hand, parallel DIRK formulas offer some advantage. For example, Iserles and N rsett [26] derive a family of 4 stage, 4 th order 2 parallel DIRK formulas, which includes the L stable (but not B stable) ....

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P. J. van der Houwen and B. P. Sommeijer, "Parallel iteration of high-order Runge-Kutta methods with stepsize control", J. Comput. Appl. Math., vol. 29, pp. 111--127, 1990.

....approximation to the matrix exponential will be accurate in those eigenvalues, thus accommodating stiffness. Note that there have been several recent efforts to design algorithms for the solution of time dependent problems, some of which may be particularly suited to parallel processing; see [19, 21, 22, 43, 48] and [49] for a review. It should also be mentioned that the approach we are using to aproximate and evaluate the exponential of the reduced operator has its roots in the work of Varga [52] and in previous work by the authors and others; see Section 3 for details. The structure of our paper is as ....

P. J. van der Houwen and B. P. Sommeijer, Parallel iteration of high-order Runge-Kutta methods with stepsize control, J. Comput. Appl. Math., 29 (1990), pp. 111--127.

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P.J. van der Houwen and B.P. Sommeijer. Parallel Iteration of high--order Runge--Kutta Methods with stepsize control. Journal of Computational and Applied Mathematics, 29:111--127, 1990.

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