Adapted explicit two-step peer methods

Conte, Dajana; D’Ambrosio, Raffaele; Moccaldi, Martina; Paternoster, Beatrice

doi:10.1515/jnma-2017-0102

Cited by 22 publications

(6 citation statements)

References 31 publications

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“…Thanks to the structure of the coefficient matrix, those methods can be easily parallelized, so the computational effort can be reduced. In the future we aim to construct such types of methods for different operators such as stochastic differential equations [ 15 , 21 , 31 ], fractional differential equations [ 2 , 10 , 13 , 16 ], partial differential equations [ 1 , 11 , 14 , 20 , 30 , 32 , 35 , 38 , 40 ], Volterra integral equations [ 8 , 9 , 12 , 17 , 23 ], second order problems [ 26 , 37 ], oscillatory problems [ 19 , 22 , 24 , 33 , 36 , 53 ], as well as to the development of algebraically stable high order collocation based multivalue methods [ 18 , 29 ].…”

Section: Discussionmentioning

confidence: 99%

Multivalue Almost Collocation Methods with Diagonal Coefficient Matrix

Conte

D’Ambrosio

D'Arienzo

et al. 2020

Computational Science and Its Applications – ICCSA 2020

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We introduce a family of multivalue almost collocation methods with diagonal coefficient matrix for the numerical solution of ordinary differential equations. The choice of this type of coefficient matrix permits a reduction of the computational cost and a parallel implementation. Collocation gives a continuous extension of the solution which is useful for a variable step size implementation. We provide examples of A-stable methods with two and three stages and order 3.

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Section: Discussionmentioning

confidence: 99%

Multivalue Almost Collocation Methods with Diagonal Coefficient Matrix

Conte

D’Ambrosio

D'Arienzo

et al. 2020

Computational Science and Its Applications – ICCSA 2020

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“…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

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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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Adapted explicit two-step peer methods

Cited by 22 publications

References 31 publications

Multivalue Almost Collocation Methods with Diagonal Coefficient Matrix

Multivalue Almost Collocation Methods with Diagonal Coefficient Matrix

Exponentially fitted methods with a local energy conservation law

Exponentially fitted methods that preserve conservation laws

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