Splitting K-symplectic methods for non-canonical separable Hamiltonian problems

2016 ◽  
Vol 322 ◽  
pp. 387-399 ◽  
Author(s):  
Beibei Zhu ◽  
Ruili Zhang ◽  
Yifa Tang ◽  
Xiongbiao Tu ◽  
Yue Zhao
Keyword(s):  
Symplectic Methods ◽  
Hamiltonian Problems

SYMPLECTIC NUMERICAL METHODS FOR HAMILTONIAN PROBLEMS

1993 ◽  
Vol 04 (02) ◽  
pp. 385-392 ◽  
Author(s):  
J. M. SANZ-SERNA ◽  
M. P. CALVO
Keyword(s):  
Numerical Methods ◽  
Numerical Integration ◽  
Numerical Experiments ◽  
Symplectic Methods ◽  
Hamiltonian Problems ◽  
Integral Invariants

We consider symplectic methods for the numerical integration of Hamiltonian problems, i.e. methods that preserve the Poincaré integral invariants. Examples of symplectic methods are given and numerical experiments are reported.

scholarly journals Efficiency of Runge-Kutta Methods in Solving Simple Harmonic Oscillators

MATEMATIKA ◽  
2018 ◽  
Vol 34 (1) ◽  
pp. 1-12
Author(s):  
Annie Gorgey ◽  
Nor Azian Aini Mat
Keyword(s):  
Harmonic Oscillator ◽  
Numerical Solutions ◽  
Harmonic Oscillators ◽  
Symplectic Method ◽  
Symplectic Methods ◽  
Error Growth ◽  
Hamiltonian Problems ◽  
Simple Harmonic Oscillator ◽  
Short Period ◽  
Gauss Method

Symmetric methods such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR) and 2-stage Gauss method are beneficial in solving Hamiltonian problems since they are also symplectic. Symplectic methods have advantages over non-symplectic methods in the long term integration of Hamiltonian problems. The study is to show the efficiency of IMR, ITR and the 2-stage Gauss method in solving simple harmonic oscillators (SHO). This study is done theoretically and numerically on the simple harmonic oscillator problem. The theoretical analysis and numerical results on SHO problem showed that the magnitude of the global error for a symmetric or symplectic method with stepsize h is linearly dependent on time t. This gives the linear error growth when a symmetric or symplectic method is applied to the simple harmonic oscillator problem. Passive and active extrapolations have been implemented to improve the accuracy of the numerical solutions. Passive extrapolation is observed to show quadratic error growth after a very short period of time. On the other hand, active extrapolation is observed to show linear error growth for a much longer period of time.

scholarly journals Adjustment of Force–Gradient Operator in Symplectic Methods

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2718
Author(s):  
Lina Zhang ◽  
Xin Wu ◽  
Enwei Liang
Keyword(s):  
Spiral Structure ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Time Reversibility ◽  
Hamiltonian Problems ◽  
Gradient Operator ◽  
Numerical Tests ◽  
Integration Algorithms ◽  
Dynamical Features ◽  
Numerical Performance

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian H=T(p)+V(q) with kinetic energy T(p)=p2/2 in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems H=K(p,q)+V(q) with integrable part K(p,q)=∑i=1n∑j=1naijpipj+∑i=1nbipi, where aij=aij(q) and bi=bi(q) are functions of coordinates q. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies.

TWO NEW PHASE-FITTED SYMPLECTIC PARTITIONED RUNGE–KUTTA METHODS

2011 ◽  
Vol 22 (12) ◽  
pp. 1343-1355 ◽  
Author(s):  
TH. MONOVASILIS ◽  
Z. KALOGIRATOU ◽  
T. E. SIMOS
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Phase Lag ◽  
Symplectic Methods ◽  
Hamiltonian Problems ◽  
Algebraic Order ◽  
Runge Kutta ◽  
New Phase

New symplectic Partitioned Runge–Kutta (SPRK) methods with phase-lag of order infinity are derived in this paper. Specifically two new symplectic methods are constructed with second and third algebraic order. The methods are tested on the numerical integration of Hamiltonian problems and on the estimation of the eigenvalues of the Schrödinger equation.

scholarly journals Conservative Galerkin methods for dispersive Hamiltonian problems

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
James Jackaman ◽  
Tristan Pryer
Keyword(s):  
Galerkin Methods ◽  
Hamiltonian Problems
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