scholarly journals Relaxation Runge–Kutta Methods for Hamiltonian Problems

2020 ◽  
Vol 84 (1) ◽  
Author(s):  
Hendrik Ranocha ◽  
David I. Ketcheson
Keyword(s):  
Hamiltonian Problems ◽  
Runge Kutta

scholarly journals On a Class of Conjugate Symplectic Hermite-Obreshkov One-Step Methods with Continuous Spline Extension

Axioms ◽  
2018 ◽  
Vol 7 (3) ◽  
pp. 58 ◽  
Author(s):  
Francesca Mazzia ◽  
Alessandra Sestini
Keyword(s):  
Differential Equation ◽  
Initial Value Problems ◽  
Multistep Methods ◽  
Maximal Order ◽  
Linear Multistep Methods ◽  
Initial Value ◽  
Hamiltonian Problems ◽  
Efficient Approach ◽  
One Step ◽  
Runge Kutta

The class of A-stable symmetric one-step Hermite–Obreshkov (HO) methods introduced by F. Loscalzo in 1968 for dealing with initial value problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points. As a new result, it is shown that these maximal order schemes are conjugate symplectic, which is a benefit when the methods have to be applied to Hamiltonian problems. Furthermore, a new efficient approach for the computation of the spline extension is introduced, adopting the same strategy developed for the BS linear multistep methods. The performances of the schemes are tested in particular on some Hamiltonian benchmarks and compared with those of the Gauss–Runge–Kutta schemes and Euler–Maclaurin formulas of the same order.

AN ADAPTED SYMPLECTIC INTEGRATOR FOR HAMILTONIAN PROBLEMS

2001 ◽  
Vol 12 (02) ◽  
pp. 225-234 ◽  
Author(s):  
JESÚS VIGO-AGUIAR ◽  
T. E. SIMOS ◽  
A. TOCINO
Keyword(s):  
Numerical Results ◽  
New Method ◽  
Symplectic Integrators ◽  
Order Method ◽  
Symplectic Integrator ◽  
Hamiltonian Problems ◽  
Trigonometric Fitting ◽  
Algebraic Order ◽  
Runge Kutta ◽  
Trigonometrically Fitted Method

In this paper, a new procedure for deriving efficient symplectic integrators for Hamiltonian problems is introduced. This procedure is based on the combination of the trigonometric fitting technique and symplecticness conditions. Based on this procedure, a simple modified Runge–Kutta–Nyström second algebraic order trigonometrically fitted method is developed. We present explicity the symplecticity conditions for the new modified Runge–Kutta–Nyström method. Numerical results indicate that the new method is much more efficient than the "classical" symplectic Runge–Kutta–Nyström second algebraic order method introduced in Ref. 1.

scholarly journals On a Class of Hermite-Obrechkoff One-Step Methods with Continuous Spline Extension

2018 ◽  
Author(s):  
Francesca Mazzia ◽  
Alessandra Sestini
Keyword(s):  
Differential Equation ◽  
Initial Value Problems ◽  
Multistep Methods ◽  
Maximal Order ◽  
Linear Multistep Methods ◽  
Initial Value ◽  
Hamiltonian Problems ◽  
Efficient Approach ◽  
One Step ◽  
Runge Kutta

The class of A-stable symmetric one-step Hermite-Obrechkoff (HO) methods introduced in [1] for dealing with Initial Value Problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points. As a new result, it is shown that these maximal order schemes are conjugate symplectic which is a benefit when the methods have to be applied to Hamiltonian problems. Furthermore a new efficient approach for the computation of the spline extension is introduced, adopting the same strategy developed in [2] for the BS linear multistep methods. The performances of the schemes are tested in particular on some Hamiltonian benchmarks and compared with those of the Gauss Runge-Kutta schemes and Euler-Maclaurin formulas of the same order.

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 A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1103
Author(s):  
Felice Iavernaro ◽  
Francesca Mazzia
Keyword(s):  
Fourth Order ◽  
New Method ◽  
Symplectic Method ◽  
Hamiltonian Problems ◽  
First Order ◽  
Runge Kutta Method ◽  
Midpoint Method ◽  
Higher Derivatives ◽  
Runge Kutta ◽  
Order Four

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.

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