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### A Hopf Bifurcation Theorem for Difference Equations Approximating a Differential Equation.

Monatshefte für Mathematik

### A modified version of explicit Runge-Kutta methods for energy-preserving

Kybernetika

In this paper, Runge-Kutta methods are discussed for numerical solutions of conservative systems. For the energy of conservative systems being as close to the initial energy as possible, a modified version of explicit Runge-Kutta methods is presented. The order of the modified Runge-Kutta method is the same as the standard Runge-Kutta method, but it is superior in energy-preserving to the standard one. Comparing the modified Runge-Kutta method with the standard Runge-Kutta method, numerical experiments...

### A Note on Multistep Methods and Attracting Sets of Dynamical Systems.

Numerische Mathematik

### A0-Stability and Stiff Stability of Brown's Multistep Multiderivative Methods.

Numerische Mathematik

### Bounds on Nonlinear Operators in Finite-dimensional Banach Spaces.

Numerische Mathematik

### Comparing numerical integration schemes for a car-following model with real-world data

Programs and Algorithms of Numerical Mathematics

A key element of microscopic traffic flow simulation is the so-called car-following model, describing the way in which a typical driver interacts with other vehicles on the road. This model is typically continuous and traffic micro-simulator updates its vehicle positions by a numerical integration scheme. While increasing the order of the scheme should lead to more accurate results, most micro-simulators employ the simplest Euler rule. In our contribution, inspired by [1], we will provide some additional...

### Delay-dependent stability of linear multi-step methods for linear neutral systems

Kybernetika

In this paper, we are concerned with numerical methods for linear neutral systems with multiple delays. For delay-dependently stable neutral systems, we ask what conditions must be imposed on linear multi-step methods in order that the numerical solutions display stability property analogous to that displayed by the exact solutions. Combining with Lagrange interpolation, linear multi-step methods can be applied to the neutral systems. Utilizing the argument principle, a sufficient condition is derived...

### Delay-dependent stability of Runge-Kutta methods for linear neutral systems with multiple delays

Kybernetika

In this paper, we are concerned with stability of numerical methods for linear neutral systems with multiple delays. Delay-dependent stability of Runge-Kutta methods is investigated, i. e., for delay-dependently stable systems, we ask what conditions must be imposed on the Runge-Kutta methods in order that the numerical solutions display stability property analogous to that displayed by the exact solutions. By means of Lagrange interpolation, Runge-Kutta methods can be applied to neutral differential...

### Erweiterung des $G$-Stabilitätsbegriffes auf die Klasse der linearen Mehrschrittblockverfahren.

Aplikace matematiky

In der vorliegenden Arbeit wird der $G$-Stabilitätsbegriff von Dahlquist, der die Grundlage für Stabilitätsuntersuchungen bei linearen Mehrschrittverfahren zur Lösung nichtlinearet Anfangswertaufgaben bildet, auf die Klasse der linearen Mehrschrittblockverfahren übertragen. Es wird nachgewiesen, das Blockverfahren, die in diesem Sinne stabil sind, höchstens die Konsistenzordnung 2 haben können.

### Explizite Konstruktion von linearen Mehrschrittblockverfahren

Aplikace matematiky

In der vorliegenden Arbeit wird für lineare Mehrschrittblock verfahren zur numerischen Lösung von Anfangswertaufgaben eine explizite Konstruktionsmöglichkeit angegeben. Sie ermöglicht es, zu einem gegebenen Stabilitätspolynom ohne Lösung eines linearen Gleichungssystems die Koefizienten des zugehörigen Blockverfahrens zu berechnen.

### Former super-irrécductibles des systèmes différentiels linéaires.

Numerische Mathematik

### Generalized periodic overimplicit multistep methods (GPOM methods)

Aplikace matematiky

The paper deals with some new methods for the numerical solution of initial value problems for ordinary differential equations. The main idea of these methods consists in the fact that in one step of the method a group of unknown values of the approximate solution is computed simultaneously. The class of methods under investigation is wide enough to contain almost all known classical methods. Sufficient conditions for convergence are found.

### Generalized Runge-Kutta Methods of Order Four with Stepsize Control for Stiff Ordinary Differential Equations.

Numerische Mathematik

### Implicit for local effects and explicit for nonlocal effects is unconditionally stable.

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

### Necessary conditions for the convergence of the generalized periodic overimplicit multistep method

Aplikace matematiky

This paper is the continuation of the paper "Generalized periodic overimplicit multistep methods" of the same author and it deals with the necessary and, in some special cases, with the necessary and sufficient conditions for the convergence of general periodic overimplicit multistep methods.

### Nichtäquidistante Diskretisierungen von Randwertaufgaben.

Numerische Mathematik

### Numerical stability in solution of ordinary differential equations

Aplikace matematiky

### Numerical stability test of neutral delay differential equations.

Mathematical Problems in Engineering

### On Clenshaws's Method and a Generalisation to Faber Series.

Numerische Mathematik

### On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D⋆⋆

Mathematical Modelling of Natural Phenomena

We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for...

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