Parallel iteration of two-step Runge-Kutta methods

2021 ◽  
Vol 66 (1) ◽  
pp. 12-24
Author(s):  
Thuy Nguyen Thu
Keyword(s):  
Iteration Process ◽  
Parallel Computers ◽  
Initial Value ◽  
Parallel Methods ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Parallel Iteration ◽  
Predictor Corrector ◽  
First Order Differential Equations

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.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
N. A. Ahmad ◽  
N. Senu ◽  
F. Ismail
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
First Order ◽  
Runge Kutta Method ◽  
Algebraic Order ◽  
Runge Kutta ◽  
The Stability ◽  
Order Four

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

scholarly journals Optimum Integration Procedure for Connectionist and Dynamic Field Equations

2021 ◽  
Vol 15 ◽  
Author(s):  
Andrés Rieznik ◽  
Rocco Di Tella ◽  
Lara Schvartzman ◽  
Andrés Babino
Keyword(s):  
Euler Method ◽  
Connectionist Model ◽  
Kutta Method ◽  
Field Equations ◽  
Single Digit ◽  
First Order ◽  
Runge Kutta Method ◽  
Dynamic Field ◽  
Runge Kutta ◽  
First Order Differential Equations

Connectionist and dynamic field models consist of a set of coupled first-order differential equations describing the evolution in time of different units. We compare three numerical methods for the integration of these equations: the Euler method, and two methods we have developed and present here: a modified version of the fourth-order Runge Kutta method, and one semi-analytical method. We apply them to solve a well-known nonlinear connectionist model of retrieval in single-digit multiplication, and show that, in many regimes, the semi-analytical and modified Runge Kutta methods outperform the Euler method, in some regimes by more than three orders of magnitude. Given the outstanding difference in execution time of the methods, and that the EM is widely used, we conclude that the researchers in the field can greatly benefit from our analysis and developed methods.

scholarly journals A numerical method for the solution of an autonomous initial value problem

2012 ◽  
Vol 28 (2) ◽  
pp. 305-312
Author(s):  
FLAVIUS PATRULESCU ◽  
Keyword(s):  
Stability Region ◽  
Numerical Approximation ◽  
Truncation Error ◽  
Interpolation Formula ◽  
Convergence Order ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta ◽  
The Stability ◽  
First Order Differential Equations

Using a known interpolation formula we introduce a class of numerical methods for approximating the solutions of scalar initial value problems for first order differential equations, which can be identified as explicit Runge-Kutta methods. We determine bounds for the local truncation error and we also compare the convergence order and the stability region with those for explicit Runge-Kutta methods, which have convergence order equal with number of stages (i.e. with 2, 3 and 4 stages). The convergence order is only two, but our methods have a larger absolute stability region than the above mentioned methods. In the last section a numerical example is provided, and the obtained numerical approximation is compared with the corresponding exact solution.

scholarly journals Resolução Numérica da equação diferencial de resfriamento de newton pelos métodos de EULER e RUNGE-KUTTA

2016 ◽  
Vol 2 (1) ◽  
pp. 10-25
Author(s):  
Andresa Pescador ◽  
Zilmara Raupp Quadros Oliveira
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Euler Method ◽  
Kutta Method ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Analytical Resolution ◽  
Important Branch ◽  
First Order Differential Equations

This article presents the first-order differential equations, which are a very important branch of mathematics as they have a wide applicability, in mathematics, as in physics, biology and economy. The objective of this study was to analyze the resolution of the equation that defines the cooling Newton's law. Verify its behavior using some applications that can be used in the classroom as an auxiliary instrument to the teacher in addressing these contents bringing answers to the questions of the students and motivating them to build their knowledge. It attempted to its resolution through two numerical methods, Euler method and Runge -Kutta method. Finally, there was a comparison of the approach of the solution given by the numerical solution with the analytical resolution whose solution is accurate.

scholarly journals The Scheme of 10th Order Implicit Runge-Kutta Method to Solve the First Order of Initial Value Problems

INSIST ◽  
2017 ◽  
Vol 1 (1) ◽  
Author(s):  
Z Bahri ◽  
L Zakaria ◽  
Syamsudhuha Syamsudhuha
Keyword(s):  
Initial Value Problems ◽  
Numerical Technique ◽  
Initial Value ◽  
First Order ◽  
Simulation Process ◽  
Runge Kutta Method ◽  
Simplifying Assumptions ◽  
Runge Kutta ◽  
Consistency Properties ◽  
Numerical Simulation Technique

Abstract—To construct a scheme of implicit Runge-Kutta methods, there are a number of coefficients that must be determined and satisfying consistency properties and Butcher’s simplifying assumptions. In this paper we provide the numerical simulation technique to obtain a scheme of 10th order Implicit Runge-Kutta (IRK10) method. For simulation process, we construct an algorithm to compute all the coefficients involved in the IRK10 scheme. The algorithm is implemented in a language programming (Turbo Pascal) to obtain all the required coefficients in the scheme. To show that our scheme works correctly, we use the scheme to solve Hénon-Heiles system.Keywords—ODEs, 10th order IRK method, numerical technique, Hénon-Heiles system

Sign in / Sign up

Export Citation Format

Share Document