Parallel Predictor-corrector Methods Research Articles

PHƯƠNG PHÁP LẶP SONG SONG RUNGE-KUTTA HAI BƯỚC

In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

Two-step-by-two-step PIRKN-type PC methods based on Gauss–Legendre collocation points for nonstiff IVPs

This paper investigates parallel predictor–corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations of the form \(\mathbf{y}''(t)=\mathbf{f}(t,\mathbf{y}(t))\). The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of \(s\)-stage collocation Gauss–Legendre Runge–Kutta–Nystrom (RKN) methods based on \(\hat{c}_1,\dots ,\hat{c}_s\) and \(2s\)-stage collocation RKN methods based on \(\hat{c}_1,\dots ,\hat{c}_s,1+\hat{c}_1,\dots ,1+\hat{c}_s\). At \(n\)th integration step, the stage values of the \(2s\)-stage collocation RKN methods evaluated at \(t_n+(1+\hat{c}_1)h,\dots ,t_n+(1+\hat{c}_s)h\) can be used as the stage values of the collocation Gauss–Legendre RKN method for \((n+2)\)th integration step. In this way we obtain the corrector methods in which the integration processes can be proceeded two-step by two-step. The resulting parallel PC iteration methods which are called two-step-by-two-step (TBT) parallel-iterated RKN-type (PIRKN-type) PC methods based on Gauss–Legendre collocation points (two-step-by-two-step PIRKNG methods or TBTPIRKNG methods) give us a faster integration process. Fixed stepsize applications of these TBTPIRKNG methods to three widely used test problems show that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RKN methods (PIRKN methods) and sequential codes ODEX2 and DOP853 available from the literature.

Two-step-by-two-step PIRK-type PC methods based on Gauss–Legendre collocation points

This paper concerns with parallel predictor–corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of s -stage collocation Gauss–Legendre Runge–Kutta (RK) methods based on c 1 , … , c s and the 2 s -stage collocation RK methods based on c 1 , … , c s , 1 + c 1 , … , 1 + c s . At n th integration step, the stage values of the 2 s -stage collocation RK methods evaluated at t n + ( 1 + c 1 ) h , … , t n + ( 1 + c s ) h can be used as the stage values of the collocation Gauss–Legendre RK method for ( n + 2 ) th integration step. By this way, we obtain the corrector methods in which the integration processes can be proceeded two-step-by-two-step. The resulting parallel PC iteration methods which are called two-step-by-two-step (TBT) parallel-iterated RK-type (PIRK-type) PC methods based on Gauss–Legendre collocation points (two-step-by-two-step PIRKG methods or TBTPIRKG methods) give us a faster integration process. Fixed step size applications of these TBTPIRKG methods to the three widely used test problems reveal that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RK methods (PIRK methods) and sequential codes ODEX, DOPRI5 and DOP853 available from the literature.

Twostep-by-twostep continuous PIRKN-type PC methods for nonstiff IVPs

In this paper, we start with the consideration of direct collocation-based Runge–Kutta-Nyström (RKN) methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y″( 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-by-twostep 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.

Recent advances in linear analysis of convergence for splittings for solving ODE problems

In the nineties, Van der Houwen et al. (see, e.g., [P.J. van der Houwen, B.P. Sommeijer, J.J. de Swart, Parallel predictor–corrector methods, J. Comput. Appl. Math. 66 (1996) 53–71; P.J. van der Houwen, J.J.B. de Swart, Triangularly implicit iteration methods for ODE-IVP solvers, SIAM J. Sci. Comput. 18 (1997) 41–55; P.J. van der Houwen, J.J.B. de Swart, Parallel linear system solvers for Runge–Kutta methods, Adv. Comput. Math. 7 (1–2) (1997) 157–181]) introduced a linear analysis of convergence for studying the properties of the iterative solution of the discrete problems generated by implicit methods for ODEs. This linear convergence analysis is here recalled and completed, in order to provide a useful quantitative tool for the analysis of splittings for solving such discrete problems. Indeed, this tool, in its complete form, has been actively used when developing the computational codes BiM and BiMD [L. Brugnano, C. Magherini, The BiM code for the numerical solution of ODEs, J. Comput. Appl. Math. 164–165 (2004) 145–158. Code available at: http://www.math.unifi.it/~brugnano/BiM/index.html; L. Brugnano, C. Magherini, F. Mugnai, Blended implicit methods for the numerical solution of DAE problems, J. Comput. Appl. Math. 189 (2006) 34–50]. Moreover, the framework is extended for the case of special second order problems. Examples of application, aimed to compare different iterative procedures, are also presented.

Twostep-by-twostep PIRK-type PC methods with continuous output formulas

This paper deals with parallel predictor–corrector (PC) iteration methods based on collocation Runge–Kutta (RK) corrector methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. At n th step, the continuous output formulas are used not only for predicting the stage values in the PC iteration methods but also for calculating the step values at ( n + 2 ) th step. In this case, the integration processes can be proceeded twostep-by-twostep. The resulting twostep-by-twostep (TBT) parallel-iterated RK-type (PIRK-type) methods with continuous output formulas (twostep-by-twostep PIRKC methods or TBTPIRKC methods) give us a faster integration process. Fixed stepsize applications of these TBTPIRKC methods to a few widely-used test problems reveal that the new PC methods are much more efficient when compared with the well-known parallel-iterated RK methods (PIRK methods), parallel-iterated RK-type PC methods with continuous output formulas (PIRKC methods) and sequential explicit RK codes DOPRI5 and DOP853 available from the literature.

Continuous parallel-iterated RKN-type PC methods for nonstiff IVPs

This paper investigates parallel predictor–corrector (PC) iteration schemes based on direct collocation Runge–Kutta–Nyström (RKN) corrector methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y ″ ( t ) = f ( t , y ( t ) ) . Consequently, the resulting parallel-iterated RKN-type PC methods are provided with continuous output formulas. The continuous numerical approximations are also used for predicting the stage values in the PC iteration processes. In this way, we obtain parallel PC methods with continuous output formulas and high-order predictors. Applications of the resulting parallel PC methods to a few widely-used test problems reveal that these new parallel PC methods are much more efficient when compared with the parallel-iterated RKN (PIRKN) methods and the sequential ODEX2 and DOPRIN codes from the literature.

PARALLEL PREDICTOR-CORRECTOR METHODS FOR THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS II

This work presents parallel multistep methods for the solution of ordinary differential equations. The characteristic of parallel computing is that there is a "front" and the computation at points ahead of the front depends only on information behind it. This requires a resetting of serial algorithms and may also lead to numerical errors and instabilities. The analysis of positive and negative aspects of parallel computing is the subject of this paper. Some of the methods presented below are uncommon in the literature on mathematical computing. Others have been elaborated for this study on the basis of the traditional Adams-Bashforth multistep methods. A performance comparison of the methods is made by numerical testing in molecular dynamics calculations. The increase of the number of processors m appears to seriously deteriorate the stability of the calculations and the use of m larger than 2 seems impractical.

A class of explicit parallel two-step Runge-Kutta methods

The aim of the present paper is to construct a class of two-step Runge-Kutta methods of arbitrarily high order for application to parallel computers. Starting with ans-stage implicit two-step Runge-Kutta method of orderp withk=p/2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P(EC) m E mode. In this way, we obtain an explicit two-step Runge-Kutta method that has orderp for allm and that requiresk(m+1) right-hand side evaluations per step of which eachk evaluation can be computed in parallel. By a number of numerical experiments we show the superiority of the parallel predictor-corrector methods proposed here over parallel method available in the literature.

Some P(EC)mE methods for the parallel solution of ODEs

Abstract Stability and efficiency are two important considerations in deriving good numerical methods for ODEs. The underlying challenge is to increase the stability region while maintaining or even improving efficiency. In this paper, we consider this problem within the context of parallel solution methods. A family of methods which we call parallel block predictor-corrector methods is proposed. Members of this family are characterized by iterated corrector evaluations carried out in two adjacent blocks. Stability properties of these methods are analyzed and compared with some existing block-based parallel predictor-corrector methods. Performance of two particular members of the family is compared with existing block-based parallel predictor-corrector methods, using solution formulas which extend over a range of integration orders and which use various numbers of processors. The results obtained from a stability analysis and from a collection of numerical experiments indicate that the proposed methods have the potential to balance stability properties and efficiency in parallel computer systems.

Explicit parallel two-step Runge-Kutta-Nyström methods

The aim of this paper is to design a class of two-step Runge-Kutta-Nyström methods of arbitrarily high order for the special second-order equation y″( t ) = f(y( t )), for use on parallel computers. Starting with an s -stage implicit two-step Runge-Kutta-Nyström method of order p with k = p 2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P ( EC ) m E mode. In this way, we obtain an explicit two-step Runge-Kutta-Nyström method that has order p for all m and that requires k ( m + 1) right-hand side evaluations per step of which each k evaluation can be computed in parallel. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

A parallel multistep predictor-corrector algorithm for solving ordinary differential equations

In this paper we give a generalized predictor-corrector algorithm for solving ordinary differential equations with specified initial values. The method uses multiple correction steps which can be carried out in parallel with a prediction step. The proposed method gives a larger stability interval compared to the existing parallel predictor-corrector methods. A method has been suggested to implement the algorithm in multiple processor systems with efficient utilization of all the processors.

Parallel Processing Algorithms for System Parameter Identification

This paper describes a parallel maximum likelihood (PML) method for system parameter identification. Parallel processing concepts are employed to accelerate convergence and reduce computation time. The PML method is based upon a square-root-free parallel quasi-Newton (PQN) method for approximating the information matrix, and parallel predictor-corrector methods for rapidly integrating ordinary differential equations. The square-root-free formulation avoids the problem of a singular information matrix that occurs in practice. In addition, the accuracy of the maximum likelihood parameter estimates and the convergence of characteristics of the EKF are improved using the square-root-free formulation. The speed-up in computation time offered by the PML method will permit very large scale system identification problems to be solved in practice.