scholarly journals Randomized Runge–Kutta method — Stability and convergence under inexact information

2021 ◽  
pp. 101554
Author(s):  
Tomasz Bochacik ◽  
Maciej Goćwin ◽  
Paweł M. Morkisz ◽  
Paweł Przybyłowicz
Keyword(s):  
Stability And Convergence ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Inexact Information ◽  
Runge Kutta

scholarly journals Accurate Numerical Method for Singular Initial-Value Problems

2021 ◽  
Vol 08 (03) ◽  
pp. 01-11
Author(s):  
Tesfaye Aga Bullo ◽  
Gemechis File Duressa ◽  
Gashu Gadisa Kiltu
Keyword(s):  
Numerical Method ◽  
Initial Value Problems ◽  
Approximate Solutions ◽  
Stability And Convergence ◽  
Order System ◽  
Kutta Method ◽  
Initial Value ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Fifth Order

In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.

scholarly journals Construction and Implementation of Optimal 8--Step Linear Multistep method

2017 ◽  
Vol 4 (1) ◽  
Author(s):  
Omolara Bakre ◽  
Gbemisola Awe ◽  
Moses Akanbi
Keyword(s):  
Taylor Expansion ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Linear Multistep Method ◽  
Kutta Method ◽  
Numerical Examples ◽  
Runge Kutta Method ◽  
Runge Kutta

In this paper, the optimal 8--step linear multistep method for solving $y^{\prime}=f(x,y)$ is constructed and implemented. The construction was carried out using the technique based on the Taylor expansion of $y(x + jh)$ and $y^{\prime}(x + jh)$ about $x + t h$, where \emph{t} need not necessarily be an integer. The consistency, stability and convergence of the proposed method are investigated. To investigate the accuracy of the method, a comparison with the classical 8-stage Runge--Kutta method is carried out on two numerical examples. The results obtained by the constructed method are accurate up to certain degrees and compete favourably with those produced by the classical 8-stage Runge--Kutta method

scholarly journals Nonlinear Stability and Convergence of Two-Step Runge-Kutta Methods for Volterra Delay Integro-Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Haiyan Yuan ◽  
Cheng Song
Keyword(s):  
Differential Equations ◽  
Quadrature Formula ◽  
Nonlinear Stability ◽  
Stability And Convergence ◽  
Asymptotically Stable ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Stage Order ◽  
Runge Kutta ◽  
The Stability

This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of(k,l)-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a(k,l)-algebraically stable two-step Runge-Kutta method with0<k<1is proved. For the convergence, the concepts ofD-convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order isp, then the method with compound quadrature formula isD-convergent of order at leastmin{p,ν}, whereνdepends on the compound quadrature formula.

Application of the piecewise constant method in nonlinear dynamics of drill string

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jialin Tian ◽  
Jie Wang ◽  
Yi Zhou ◽  
Lin Yang ◽  
Changyue Fan ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamic Model ◽  
Dynamic Characteristics ◽  
Drill String ◽  
Vibration Velocity ◽  
Kutta Method ◽  
Time Step ◽  
Piecewise Constant ◽  
Runge Kutta Method ◽  
Runge Kutta

Abstract Aiming at the current development of drilling technology and the deepening of oil and gas exploration, we focus on better studying the nonlinear dynamic characteristics of the drill string under complex working conditions and knowing the real movement of the drill string during drilling. This paper firstly combines the actual situation of the well to establish the dynamic model of the horizontal drill string, and analyzes the dynamic characteristics, giving the expression of the force of each part of the model. Secondly, it introduces the piecewise constant method (simply known as PT method), and gives the solution equation. Then according to the basic parameters, the axial vibration displacement and vibration velocity at the test points are solved by the PT method and the Runge–Kutta method, respectively, and the phase diagram, the Poincare map, and the spectrogram are obtained. The results obtained by the two methods are compared and analyzed. Finally, the relevant experimental tests are carried out. It shows that the results of the dynamic model of the horizontal drill string are basically consistent with the results obtained by the actual test, which verifies the validity of the dynamic model and the correctness of the calculated results. When solving the drill string nonlinear dynamics, the results of the PT method is closer to the theoretical solution than that of the Runge–Kutta method with the same order and time step. And the PT method is better than the Runge–Kutta method with the same order in smoothness and continuity in solving the drill string nonlinear dynamics.

Sign in / Sign up

Export Citation Format

Share Document