An Optimal Runge–Kutta Method for Steady-State Solutions of Hyperbolic Systems

1992 ◽  
Vol 29 (2) ◽  
pp. 425-438 ◽  
Author(s):  
Chichia Chiu ◽  
David A. Kopriva
Keyword(s):  
Steady State ◽  
Hyperbolic Systems ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Steady State Solutions

A Fast Computational Approach in Optimal Distributed-Parameter State Estimation

1983 ◽  
Vol 105 (1) ◽  
pp. 1-10 ◽  
Author(s):  
K. Watanabe ◽  
M. Iwasaki
Keyword(s):  
Steady State ◽  
Riccati Equation ◽  
Computation Time ◽  
Computational Approach ◽  
Distributed Parameter ◽  
Kutta Method ◽  
Step Size ◽  
Runge Kutta Method ◽  
Operator Riccati Equation ◽  
Runge Kutta

A fast computational approach is considered for solving of a time-invariant operator Riccati equation accompanied with the optimal steady-state filtering problem of a distributed-parameter system. The partitioned filter with the effective initialization is briefly explained and some relationships between its filter and the well-known Kalman-type filter are shown in terms of the Meditch-type fixed-point smoother in Hilbert spaces. Then, with the aid of these results the time doubling algorithm is proposed to solve the steady-state solution of the operator Riccati equation. Some numerical examples are included and a comparison of the computation time required by the proposed method is made with other algorithms—the distributed partitioned numerical algorithm, and the Runge-Kutta method. It is found that the proposed algorithm is approximately from 40 to 50 times faster than the classical Runge-Kutta method with constant step-size for the case of 9th order mode Fourier expansion.

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.

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