The existence of explicit symplectic ARKN methods with several stages and algebraic order greater than two

2019 ◽  
Vol 353 ◽  
pp. 204-209 ◽  
Author(s):  
Jiyong Li ◽  
Wei Shi ◽  
Xinyuan Wu
Keyword(s):  
Algebraic Order

Hybrid high algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives

2016 ◽  
Vol 27 (05) ◽  
pp. 1650049 ◽  
Author(s):  
Junyan Ma ◽  
T. E. Simos
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Truncation Error ◽  
Resonance Problem ◽  
Phase Lag ◽  
Step Method ◽  
Algebraic Order ◽  
Test Equation ◽  
The Stability ◽  
Scalar Test Equation

A hybrid tenth algebraic order two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives are obtained in this paper. We will investigate • the construction of the method • the local truncation error (LTE) of the newly obtained method. We will also compare the lte of the newly developed method with other methods in the literature (this is called the comparative LTE analysis) • the stability (interval of periodicity) of the produced method using frequency for the scalar test equation different from the frequency used in the scalar test equation for phase-lag analysis (this is called stability analysis) • the application of the newly obtained method to the resonance problem of the Schrödinger equation. We will compare its effectiveness with the efficiency of other known methods in the literature. It will be proved that the developed method is effective for the approximate solution of the Schrödinger equation and related periodical or oscillatory initial value or boundary value problems.

scholarly journals Bicritical Central Point of Ising Model Phase Diagram

2010 ◽  
Vol 2010 ◽  
pp. 1-5 ◽  
Author(s):  
Yahia Boughaleb ◽  
Mohammed Nouredine ◽  
Mohamed Snina ◽  
Rachid Nassif ◽  
Mohamed Bennai
Keyword(s):  
Phase Diagram ◽  
Order Parameter ◽  
Present Model ◽  
Central Point ◽  
Square Lattice ◽  
Ordered Structures ◽  
Ordered Phases ◽  
Algebraic Order ◽  
Pure Phases ◽  
Model Phase

We deal with a 2D half occupied square lattice with repulsive interactions between first and second neighboring particles. Despite the intensive studies of the present model the central point of the phase diagram for which the ratio of the two interaction strengths is still open. In the present paper we show, using standard Monte Carlo calculations, that the situation corresponds to a phase of mixed ordered structures quantified by an “algebraic” order parameter defined as the sum of densities of the existing ordered clusters. The introduced grandeur also characterizes the transitions towards the known pure ordered phases for the other values of as mentioned by the agreement of our results with those of the literature. The computation of the Cowley short range order parameter against suggests that the central point is bicritical and is a state to cross when passing between the two pure phases.

scholarly journals New Stable Closed Newton-Cotes Trigonometrically Fitted Formulae for Long-Time Integration

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
T. E. Simos
Keyword(s):  
Time Integration ◽  
Differential Method ◽  
Algebraic Order ◽  
Long Time ◽  
Long Time Integration ◽  
Differential Methods

The closed Newton-Cotes differential methods of high algebraic order for small number of function evaluations are unstable. In this work, we propose a new closed Newton-Cotes trigonometrically fitted differential method of high algebraic order which gives much more efficient results than the well-know ones.

scholarly journals Pointwise and Uniform Convergence of Fourier Extensions

2019 ◽  
Vol 52 (1) ◽  
pp. 139-175
Author(s):  
Marcus Webb ◽  
Vincent Coppé ◽  
Daan Huybrechs
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Gibbs Phenomenon ◽  
Lebesgue Function ◽  
Bernstein Type ◽  
Legendre Series ◽  
The Best Approximation ◽  
Open Questions ◽  
Algebraic Order

AbstractFourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

scholarly journals Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4)=f(x,u,u′,u′′)

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 246 ◽  
Author(s):  
Nizam Ghawadri ◽  
Norazak Senu ◽  
Firas Fawzi ◽  
Fudziah Ismail ◽  
Zarina Ibrahim
Keyword(s):  
Real Life ◽  
Order Ordinary Differential Equation ◽  
New Techniques ◽  
First Order ◽  
New Methods ◽  
Life Problems ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Order Four ◽  
Primary Contribution

The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u ′ , u ′ ′ ) and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed.

Sign in / Sign up

Export Citation Format

Share Document