scholarly journals Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 935 ◽  
Author(s):  
Simone Fiori
Keyword(s):  
Lie Groups ◽  
Conservative System ◽  
Euler Method ◽  
Model Formulation ◽  
Dynamical Equations ◽  
System Equation ◽  
D’Alembert Principle ◽  
Rotation Groups ◽  
Forward Euler ◽  
Forward Euler Method

The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illustrates applications of the generalized Euler–Poincaré equations on the rotation groups to a gyrostat satellite and a quadcopter drone. The numerical solution of the dynamical equations on the rotation groups is tackled via a generalized forward Euler method and an explicit Runge–Kutta integration method tailored to Lie groups.

scholarly journals Error estimates for forward Euler shock capturing finite element approximations of the one-dimensional Burgers' equation

2015 ◽  
Vol 25 (11) ◽  
pp. 2015-2042
Author(s):  
Erik Burman
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Burgers Equation ◽  
Euler Method ◽  
Second Order ◽  
Shock Capturing ◽  
Discrete Solution ◽  
A Priori Bound ◽  
Forward Euler ◽  
Forward Euler Method

We propose an error analysis for a shock capturing finite element method for the Burgers' equation using the duality theory due to Tadmor. The estimates use a one-sided Lipschitz stability (Lip+-stability) estimate on the discrete solution and are obtained in a weak norm, but thanks to a total variation a priori bound on the discrete solution and an interpolation inequality, error estimates in Lp-norms (1 ≤ p < ∞) are deduced. Both first-order artificial viscosity and a nonlinear shock capturing term that formally is of second order are considered. For the discretization in time we use the forward Euler method. In the numerical section we verify the convergence order of the nonlinear scheme using the forward Euler method and a second-order strong stability preserving Runge–Kutta method. We also study the Lip+-stability property numerically and give some examples of when it holds strictly and when it is violated.

scholarly journals PHASE SPACE ERROR CONTROL WITH VARIABLE TIME-STEPPING ALGORITHMS APPLIED TO THE FORWARD EULER METHOD FOR AUTONOMOUS DYNAMICAL SYSTEMS

2019 ◽  
pp. 16-26
Author(s):  
R. Vigneswaran ◽  
S. Thilaganathan
Keyword(s):  
Fixed Point ◽  
Phase Space ◽  
Dynamical Systems ◽  
Linear System ◽  
Error Control ◽  
Coefficient Matrix ◽  
Euler Method ◽  
Space Error ◽  
Forward Euler ◽  
Forward Euler Method

We consider a phase space stability error control for numerical simulation of dynamical systems. Standard adaptive algorithm used to solve the linear systems perform well during the finite time of integration with fixed initial condition and performs poorly in three areas. To overcome the difficulties faced the Phase Space Error control criterion was introduced. A new error control was introduced by R. Vigneswaran and Tony Humbries which is generalization of the error control first proposed by some other researchers. For linear systems with a stable hyperbolic fixed point, this error control gives a numerical solution which is forced to converge to the fixed point. In earlier, it was analyzed only for forward Euler method applied to the linear system whose coefficient matrix has real negative eigenvalues. In this paper we analyze forward Euler method applied to the linear system whose coefficient matrix has complex eigenvalues with negative large real parts. Some theoretical results are obtained and numerical results are given.

scholarly journals Remarks on Numerical Experiments of the Allen-Cahn Equations with Constraint via Yosida Approximation

2016 ◽  
Vol 2016 ◽  
pp. 1-16
Author(s):  
Tomoyuki Suzuki ◽  
Keisuke Takasao ◽  
Noriaki Yamazaki
Keyword(s):  
Numerical Experiments ◽  
Closed Interval ◽  
Euler Method ◽  
Multivalued Function ◽  
Original Model ◽  
Yosida Approximation ◽  
One Dimensional ◽  
Cahn Equation ◽  
Forward Euler ◽  
Forward Euler Method

We consider a one-dimensional Allen-Cahn equation with constraint from the viewpoint of numerical analysis. The constraint is provided by the subdifferential of the indicator function on the closed interval, which is the multivalued function. Therefore, it is very difficult to perform a numerical experiment of our equation. In this paper we approximate the constraint by the Yosida approximation. Then, we study the approximating system of the original model numerically. In particular, we give the criteria for the standard forward Euler method to give the stable numerical experiments of the approximating equation. Moreover, we provide the numerical experiments of the approximating equation.

scholarly journals The Bifurcation of Two Invariant Closed Curves in a Discrete Model

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Yingying Zhang ◽  
Yicang Zhou
Keyword(s):  
Discrete Model ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Euler Method ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Dynamical Behaviors ◽  
Discrete Population ◽  
Forward Euler ◽  
Forward Euler Method

A discrete population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

Bayesian Fusion on Lie Groups

2011 ◽  
Vol 2 (1) ◽  
Author(s):  
Kevin C. Wolfe ◽  
Michael Mashner
Keyword(s):  
Lie Groups ◽  
Molecular Spectroscopy ◽  
Rotation Group ◽  
Multivariate Statistical ◽  
Satellite Attitude ◽  
Group Data ◽  
Bayesian Fusion ◽  
Probability Densities ◽  
Euclidean Setting ◽  
Rotation Groups

An increasing number of real-world problems involve the measurement of data, andthe computation of estimates, on Lie groups. Moreover, establishing confidence in the resultingestimates is important. This paper therefore seeks to contribute to a larger theoretical frameworkthat generalizes classical multivariate statistical analysis from Euclidean space to the setting of Liegroups. The particular focus here is on extending Bayesian fusion, based on exponential familiesof probability densities, from the Euclidean setting to Lie groups. The definition and properties ofa new kind of Gaussian distribution for connected unimodular Lie groups are articulated, and ex-plicit formulas and algorithms are given for finding the mean and covariance of the fusion modelbased on the means and covariances of the constituent probability densities. The Lie groups thatfind the most applications in engineering are rotation groups and groups of rigid-body motions.Orientational (rotation-group) data and associated algorithms for estimation arise in problemsincluding satellite attitude, molecular spectroscopy, and global geological studies. In robotics andmanufacturing, quantifying errors in the position and orientation of tools and parts are impor-tant for task performance and quality control. Developing a general way to handle problemson Lie groups can be applied to all of these problems. In particular, we study the issue of howto ‘fuse’ two such Gaussians and how to obtain a new Gaussian of the same form that is ‘closeto’ the fused density.This is done at two levels of approximation that result from truncating theBaker-Campbell-Hausdorff formula with different numbers of terms. Algorithms are developedand numerical results are presented that are shown to generate the equivalent fused density withgood accuracy

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