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.

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.

scholarly journals Trigonometrically-Fitted Methods: A Review

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1197
Author(s):  
Changbum Chun ◽  
Beny Neta
Keyword(s):  
Initial Value Problems ◽  
Polynomial Interpolation ◽  
Multistep Methods ◽  
Error Reduction ◽  
Trigonometric Polynomials ◽  
Linear Multistep Methods ◽  
Initial Value ◽  
First Order ◽  
The Subject ◽  
Trigonometrically Fitted Methods

Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea of using information on the frequency of a solution to modify linear multistep methods by allowing the coefficients to depend on the frequency. Thus the methods integrate exactly appropriate trigonometric polynomials. This was done for both first order systems and second order initial value problems. Gautschi concluded that “the error reduction is not very substantial unless” the frequency estimate is close enough. As a result, no other work was done in this direction until 1984 when Neta and Ford showed that “Nyström’s and Milne-Simpson’s type methods for systems of first order initial value problems are not sensitive to changes in frequency”. This opened the flood gates and since then there have been many papers on the subject.

scholarly journals A Class of Two-Derivative Two-Step Runge-Kutta Methods for Non-Stiff ODEs

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
I. B. Aiguobasimwin ◽  
R. I. Okuonghae
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Initial Value Problems ◽  
Initial Value ◽  
Large Interval ◽  
New Class ◽  
Second Derivatives ◽  
Stiff Odes ◽  
Runge Kutta ◽  
Special Case

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

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