Parallel algorithms for initial value problems: Parallel shooting

doi:10.1016/0167-8191(92)90006-S

Parallel Computing

Volume 18, Issue 6, June 1992, Pages 661-673

https://doi.org/10.1016/0167-8191(92)90006-S Get rights and content

Abstract

The purpose of this paper is to consider the implementation of the parallel shooting for solving IVPs using a multiprocessor computing system. The paper shows the significance of these methods and demonstrates their effectiveness in controlling the numerical stability of the initial value algorithms.

References (11)

C.T.H Baker et al.
The Numerical solution of Nonlinear Problems
(1981)
C.D Conte et al.
Elementary Numerical Analysis, an Algorithmic Approach
(1981)
G Dahlquist
Stability and error bounds in the numerical integration of ordinary differential equations
Trans. Roy. Inst. Tech. Stockholm
(1959)
G Dahlquist
A special stability problems for linear multistep methods
BIT
(1963)
G Dahlquist
Stability questions for some numerical methods for ordinary differential equations

There are more references available in the full text version of this article.

Cited by (10)

On a parallel mesh-chopping algorithm for a class of initial value problems using fourth-order explicit Runge-Kutta method
2003, Applied Mathematics and Computation
In this paper, we extend the results of [Comput. Math. Appl. 35 (1998) 121] to the class of initial value problems $y ̂^{′} = f ̂ (x, y ̂)$ , $y ̂ (x_{0})= η ̂_{0}$ , represented as a system of first order differential equations. We propose a simple modification in the existing fourth-order Runge-Kutta method which makes it amenable to parallelization on p processors. A numerical example illustrating the theory is presented and theoretical estimates for speed up are given which are in agreement.
Parallel multiple shooting for the solution of initial value problems
1994, Parallel Computing
The computing time for the numerical solution of initial-value problems is closely related to the number of evaluations of the right-hand side. In general this number can only be reduced slightly on parallel computers, even if simultaneous evaluations of right-hand side are counted as one evaluation.
For special problems, however, it is possible to construct special methods which show a remarkable speedup on parallel computers. Multiple shooting, a method for boundary-value problems with an inherent parallelism, can also be applied efficiently to linear initial-value problems and to non-linear initial-value problems if good approximations are available.
Methods based on boundary value techniques for solving parabolic equations on parallel computers
1993, Parallel Computing
In this paper boundary value techniques for solving parabolic equations (PBV methods) will be proposed. The methods will be derived by the BVM methods for ODEs studied in [11, 12], the parallel implementation of which seems to be particularly efficient, especially for differential systems with a steady-state solution. The stability and the convergence properties of the proposed PBV methods will be studied. Numerical tests will be given both to illustrate the numerical features and the performance of the parallel version of some PBV method on a network of transputers. In particular we will consider the parallel implementation of the PBV method based on the Simpson rule which will be compared with its scalar version and with the scalar Hindmarsch's LSODE code.
Parallel modeling of transient states analysis in electrical circuits
2005, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
A new approach to the speculative method for the transient states analysis
2005, Proceedings of SPIE - The International Society for Optical Engineering
Scalable parallel RK solvers for ODEs derived by the method of lines
2004, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

View all citing articles on Scopus

View full text

PaperParallel algorithms for initial value problems: Parallel shooting

Abstract

The Numerical solution of Nonlinear Problems

Elementary Numerical Analysis, an Algorithmic Approach

Stability and error bounds in the numerical integration of ordinary differential equations

Trans. Roy. Inst. Tech. Stockholm

A special stability problems for linear multistep methods

BIT

Stability questions for some numerical methods for ordinary differential equations

Paper
Parallel algorithms for initial value problems: Parallel shooting