scholarly journals Simulation of ordinary differential equations on manifolds: some numerical experiments and verifications

Author(s):  
Arne Marthinsen ◽  
Hans Munthe-Kaas ◽  
Brynjulf Owren
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.


2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.


Author(s):  
Kamil R. Aida-zade ◽  
Vagif M. Abdullayev

AbstractThis work is dedicated to the numerical solution of a class of parametric identification problems for dynamic objects. The process is described by a system of loaded ordinary differential equations. Observations over the object have integral (interval) and point characters, at which the results of the observations are given in a summarized non-separated form. We propose a technique that reduces the solution of the initial problem to the solution of specially-built supplementary Cauchy problems. The results of some numerical experiments are also given.


2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Lee Ken Yap ◽  
Fudziah Ismail

The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.


2020 ◽  
Vol 51 (2) ◽  
pp. 123-136
Author(s):  
Iskandar Shah Mohd Zawawi

In this paper, the block backward differentiation α formulas (BBDF-α) is derived for solving first order stiff ordinary differential equations with oscillating solutions. The consistency and zero stability conditions are investigated to prove the convergence of the method. The stability region in the entire negative half plane shows that the derived method is A-stable for certain values of α. The implementation of the method using Newton iteration is also discussed. Several numerical experiments are conducted to demonstrate the performance of the method in terms of accuracy and computational time.


Author(s):  
P. Tumba ◽  
J. Sabo ◽  
A. A. Okeke ◽  
D. I. Yakoko

The new accurate implicit quarter step first derivative blocks hybrid method for solving ordinary differential equations have been proposed in this paper via interpolation and collocation method for the solution of stiff ODEs. The analysis of the method was study and it was found to be consistent, convergent, zero-stability, We further compute the region of absolute stability region and it was found to be Aα − stable . It is obvious that, the numerical experiments considered showed that the methods compete favorably with existing ones. Thus, the pair of numerical methods developed in this research is computationally reliable in solving first order initial value problems, as the results from numerical solutions of stiff ODEs shows that this method is superior and best to solve such problems as in tables and figures.


Author(s):  
Yu Ying

Runge-Kutta schemes play a very important role in solving ordinary differential equations numerically. At first we want to present the Sage routine for calculation of Butcher matrix, we call it an rk package. We tested our Sage routine in several numerical experiments with standard and symplectic schemes and verified our result by corporation with results of the calculations made by hand.Second, in Sage there are the excellent tools for investigation of algebraic sets, based on Gröbner basis technique. As we all known, the choice of parameters in Runge- Kutta scheme is free. By the help of these tools we study the algebraic properties of the manifolds in affine space, coordinates of whose are Butcher coefficients in Runge-Kutta scheme. Results are given both for explicit Runge-Kutta scheme and implicit Runge-Kutta scheme by using our rk package. Examples are carried out to justify our results. All calculation are executed in the computer algebra system Sage.


Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 185
Author(s):  
Angelamaria Cardone ◽  
Dajana Conte ◽  
Raffaele D’Ambrosio ◽  
Beatrice Paternoster

The present paper illustrates some classes of multivalue methods for the numerical solution of ordinary and fractional differential equations. In particular, it focuses on two-step and mixed collocation methods, Nordsieck GLM collocation methods for ordinary differential equations, and on two-step spline collocation methods for fractional differential equations. The construction of the methods together with the convergence and stability analysis are reported and some numerical experiments are carried out to show the efficiency of the proposed methods.


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