Exponentially fitted two-step peer methods for oscillatory problems

Conte, Dajana; Mohammadi, Fakhrodin; Moradi, Leila; Paternoster, Beatrice

doi:10.1007/s40314-020-01202-x

Cited by 16 publications

(4 citation statements)

References 52 publications

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“…To obtain suitable values of r, we specifically considered the optimization of the local truncation error of the primary formula in (8). To get the local truncation error, we expand in Taylor series about x n the formula (8) as follows:…”

Section: Mathematical Formulationmentioning

confidence: 99%

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A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

Abdulganiy

Qureshi

Soomro

et al. 2022

Preprint

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There are numerous models in the form of ordinary differential equations that require numerical treatment, for they have no closed-form solutions due to some reasons, including non-linearity, stiffness, singularity, and stability. Having in mind the increasing demand for efficient and cost-effective algorithms yet straightforward to code; an attempt is made in this research paper to introduce a new time-efficient optimal exponentially fitted numerical algorithm whose qualitative analysis unfolds its local truncation error, zero-stability, A-stability, consistency, convergence, and accuracy (at least third-order). The proposed algorithm is also implemented in an adaptive step-size mode, whereas both constant and adaptive approaches outperform several existing algorithms when tested upon differential models taken from applied sciences. The efficiency curves further confirm the time-effectiveness of the proposed exponentially fitted one-step optimal method.

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Section: Mathematical Formulationmentioning

confidence: 99%

“…In general, exponentially fitted and adapted methods can provide a precise solution by getting a priori knowledge of the solution's qualitative characteristics. Traditional numerical integrators may require a minimum step size to follow the oscillations, especially as the frequency increases ( [8]).…”

Section: Introductionmentioning

confidence: 99%

A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

Abdulganiy

Qureshi

Soomro

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…However, when unknown, the frequency can be estimated by using one of the many approaches suggested in literature [18,42,43]. Exponential fitting techniques have been successfully used to solve problems of very different nature, such as fractional differential equations [7], quadrature [14,16,24], interpolation [21], time and space integrators for ODEs [12,15,41] and PDEs [8,13,19], integral equations [9,10], boundary value problems [32].…”

Section: Introductionmentioning

confidence: 99%

Exponentially fitted methods with a local energy conservation law

Conte¹,

Frasca-Caccia²

2022

Preprint

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A new exponentially fitted version of the Discrete Variational Derivative method for the efficient solution of oscillatory complex Hamiltonian Partial Differential Equations is proposed. When applied to the nonlinear Schrödinger equation, the new scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem whose solution is an oscillatory breather wave.

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“…Exponential fitting techniques have been used in several contexts, such as fractional differential equations [4], quadrature [9,14,16,26,27], interpolation [24], peer integrators for ODEs and PDEs [13,15], integral equations [8], boundary value problems [36].…”

Section: Introductionmentioning

confidence: 99%

Exponentially fitted methods that preserve conservation laws

Conte,

Frasca-Caccia

2021

Preprint

Self Cite

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The exponential fitting technique uses information on the expected behaviour of the solution of a differential problem to define accurate and efficient numerical methods. In particular, exponentially fitted methods are very effective when applied to problems with oscillatory solutions. In this cases, compared to standard methods, they have proved to be very accurate even using large integration steps.In this paper we consider exponentially fitted Runge-Kutta methods and we give characterizations of those that preserve local conservation laws of linear and quadratic quantities.As benchmark problems we consider wave equations arising as models in several fields such as fluid dynamics and quantum physics, and derive exponentially fitted methods that preserve their conservation laws of mass (or charge) and momentum. The proposed methods are applied to approximate breather wave solutions and are compared to other known methods of the same order.

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Exponentially fitted two-step peer methods for oscillatory problems

Cited by 16 publications

References 52 publications

A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

A Novel Optimized Exponentially Fitted Algorithm with Time-Efficiency

Exponentially fitted methods with a local energy conservation law

Exponentially fitted methods that preserve conservation laws

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