Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems

Junaid Ahmad; Yousaf Habib; Shafiq Rehman; Azqa Arif; Saba Shafiq; Muhammad Younas

doi:10.3390/sym11020142

Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems

Symmetry ◽

10.3390/sym11020142 ◽

2019 ◽

Vol 11 (2) ◽

pp. 142 ◽

Cited By ~ 1

Author(s):

Junaid Ahmad ◽

Yousaf Habib ◽

Shafiq Rehman ◽

Azqa Arif ◽

Saba Shafiq ◽

...

Keyword(s):

Hamiltonian System ◽

Numerical Methods ◽

Energy Conservation ◽

Hamiltonian Systems ◽

Numerical Experiments ◽

Effective Order ◽

Good Energy ◽

Runge Kutta ◽

Separable Hamiltonian System ◽

Selection Of

A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system confirming the efficiency of the approach is also provided with good energy conservation.

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A Class of New Pouzet-Runge-Kutta-Type Methods for Nonlinear Functional Integro-Differential Equations

Abstract and Applied Analysis ◽

10.1155/2012/642318 ◽

2012 ◽

Vol 2012 ◽

pp. 1-21 ◽

Cited By ~ 3

Author(s):

Chengjian Zhang

Keyword(s):

Numerical Methods ◽

Differential Equations ◽

Lipschitz Condition ◽

Numerical Stability ◽

Numerical Experiments ◽

Integrodifferential Equations ◽

Stability Criteria ◽

Nonlinear Functional ◽

Runge Kutta ◽

Theoretical Results

This paper presents a class of new numerical methods for nonlinear functional-integrodifferential equations, which are derived by an adaptation of Pouzet-Runge-Kutta methods originally introduced for standard Volterra integrodifferential equations. Based on the nonclassical Lipschitz condition, analytical and numerical stability is studied and some novel stability criteria are obtained. Numerical experiments further illustrate the theoretical results and the effectiveness of the methods. In the end, a comparison between the presented methods and the existed related methods is given.

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Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order

The Scientific World JOURNAL ◽

10.1155/2014/147801 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Y. H. Cong ◽

C. X. Jiang

Keyword(s):

Numerical Integration ◽

Hamiltonian Systems ◽

Numerical Experiments ◽

Kutta Method ◽

Runge Kutta Method ◽

Oscillating Solutions ◽

Dispersion Order ◽

Algebraic Order ◽

Runge Kutta

The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.

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Optimization of the Navy’s three-dimensional mine impact burial prediction simulation model, Impact35, using high-order numerical methods

The Journal of Defense Modeling and Simulation Applications Methodology Technology ◽

10.1177/15485129211028661 ◽

2021 ◽

pp. 154851292110286

Author(s):

Athanasios Donas ◽

Ioannis Famelis ◽

Peter C Chu ◽

George Galanis

Keyword(s):

Numerical Analysis ◽

Numerical Methods ◽

Prediction Model ◽

Simulation Model ◽

Three Dimensional ◽

Simulation System ◽

High Order ◽

Alternative Methodology ◽

Runge Kutta ◽

Fifth Order

The aim of this paper is to present an application of high-order numerical analysis methods to a simulation system that models the movement of a cylindrical-shaped object (mine, projectile, etc.) in a marine environment and in general in fluids with important applications in Naval operations. More specifically, an alternative methodology is proposed for the dynamics of the Navy’s three-dimensional mine impact burial prediction model, Impact35/vortex, based on the Dormand–Prince Runge–Kutta fifth-order and the singly diagonally implicit Runge–Kutta fifth-order methods. The main aim is to improve the time efficiency of the system, while keeping the deviation levels of the final results, derived from the standard and the proposed methodology, low.

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Numerical methods for the compressible Navier-Stokes equations using an explicit time-marching technique (Comparison between the four stage Runge-Kutta and two stage Rational Runge-Kutta schemes)

TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series B ◽

10.1299/kikaib.52.3874 ◽

1986 ◽

Vol 52 (484) ◽

pp. 3874-3879

Author(s):

Masato FURUKAWA ◽

Hidetoshi TOMIOKA ◽

Masahiro INOUE

Keyword(s):

Numerical Methods ◽

Stokes Equations ◽

Navier Stokes ◽

Two Stage ◽

Navier Stokes Equations ◽

Time Marching ◽

Runge Kutta ◽

Technique Comparison

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Coupled systems of two-dimensional turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.422 ◽

2013 ◽

Vol 732 ◽

Author(s):

Rick Salmon

Keyword(s):

Hamiltonian System ◽

Energy Conservation ◽

Strong Correlation ◽

Conservation Law ◽

Hamiltonian Dynamics ◽

Coupled Systems ◽

The Other ◽

Two Dimensional ◽

Coherent Vortices ◽

Fluid Particles

AbstractOrdinary two-dimensional turbulence corresponds to a Hamiltonian dynamics that conserves energy and the vorticity on fluid particles. This paper considers coupled systems of two-dimensional turbulence with three distinct governing dynamics. One is a Hamiltonian dynamics that conserves the vorticity on fluid particles and a quantity analogous to the energy that causes the system members to develop a strong correlation in velocity. The other two dynamics considered are non-Hamiltonian. One conserves the vorticity on particles but has no conservation law analogous to energy conservation; the other conserves energy and enstrophy but it does not conserve the vorticity on fluid particles. The coupled Hamiltonian system behaves like two-dimensional turbulence, even to the extent of forming isolated coherent vortices. The other two dynamics behave very differently, but the behaviours of all four dynamics are accurately predicted by the methods of equilibrium statistical mechanics.

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Multi-value Numerical Methods for Hamiltonian Systems

Lecture Notes in Computational Science and Engineering - Numerical Mathematics and Advanced Applications - ENUMATH 2013 ◽

10.1007/978-3-319-10705-9_18 ◽

2014 ◽

pp. 185-193 ◽

Cited By ~ 1

Author(s):

Raffaele D’Ambrosio

Keyword(s):

Numerical Methods ◽

Hamiltonian Systems

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Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

Sains Malaysiana ◽

10.17576/jsm-2021-5006-25 ◽

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.

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