scholarly journals Variational Partitioned Runge–Kutta Methods for Lagrangians Linear in Velocities

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 861
Author(s):  
Tomasz M. Tyranowski ◽  
Mathieu Desbrun
Keyword(s):  
Algebraic System ◽  
Equations Of Motion ◽  
Variational Integrators ◽  
Lagrange Equations ◽  
First Order ◽  
Runge Kutta Method ◽  
Constant Structure ◽  
Differential Algebraic System ◽  
Main Observation ◽  
Runge Kutta

In this paper, we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the “Hamiltonian” equations of motion can be formulated as an index-1 differential-algebraic system. We also construct variational Runge–Kutta methods and analyze their properties. The general properties of Runge–Kutta methods depend on the “velocity” part of the Lagrangian. If the “velocity” part is also linear in the position coordinate, then we show that non-partitioned variational Runge–Kutta methods are equivalent to integration of the corresponding first-order Euler–Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge–Kutta method are retained. If the “velocity” part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We verified our results through numerical experiments for various dynamical systems.

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.

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.

Vibration of a Crane System Superimposed upon its Nominal Motion

2012 ◽  
Vol 430-432 ◽  
pp. 1847-1850
Author(s):  
Jin Fu Zhang ◽  
Qi Ren Luo
Keyword(s):  
Fourth Order ◽  
Equations Of Motion ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Actual Motion ◽  
Runge Kutta ◽  
The Relationship

The equations of motion for a crane system with considering the elasticity of the hoisting cable are derived. Using such equations and the relationship between the actual motion and the nominal motion of the crane system, the equations of vibration of the crane system superimposed upon its nominal motion are established. The responses of the vibration can be determined by numerically integrating the equations using the fourth order Runge–Kutta method. Based on the analysis of responses of the vibration, some conclusions concerning the vibration are obtained.

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.

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