The stability of rational approximations of cosine functions on Hilbert spaces

Alonso‐Mallo, Isaías; Cano, Begoña; Moreta, M. J.

doi:10.1016/j.apnum.2007.11.022

Cited by 7 publications

(17 citation statements)

References 18 publications

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“…In a similar way, if many steps are performed without output, only three evaluations of ψ [1] and ψ [2] are required per time step.…”

Section: Time Integrationmentioning

confidence: 99%

“…In order to obtain an explicit method with intermediate problems with simple exponentials, we consider an exponential splitting method based in two steps similar to the one dimensional case in Section 3. ψ [1] k :…”

Section: Time Integrationmentioning

confidence: 99%

“…If many steps are performed without output, only three evaluation of ψ [1] and ψ [2] are required per time step and then only 33NM + 114P M + O(N) + O(M) + O(P ) products for every step in time are necessary. Now, we study the efficiency of the splitting scheme by comparing with the fourth-order four-stage Runge-Kutta method.…”

Section: Analysis Of the Efficiency Of The Algorithmmentioning

confidence: 99%

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Time exponential splitting integrator for the Klein–Gordon equation with free parameters in the Hagstrom–Warburton absorbing boundary conditions

Alonso‐Mallo

Portillo

2018

Journal of Computational and Applied Mathematics

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The Klein-Gordon equation on an infinite two dimensional strip is considered. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton (H-W) absorbing boundary conditions (ABCs) with free parameters in the formulation of the auxiliary variables. The spatial discretization is achieved by using fourth order finite differences and the time integration is made by means of an efficient and easy to implement fourth order exponential splitting scheme which was used in [5] considering the fixed Padé parameters in the formulation of the ABCs. Here, we generalize the splitting time technique to other choices of the parameters. To check the time integrator we consider, on one hand, four types of fixed parameters, the Newmann's parameters, the Chebyshev's parameters, the Padé's parameters and optimal parameters proposed in [13] and, on the other hand, an adaptive scheme for the dynamic control of the order of absorption and the parameters. We study the efficiency of the splitting scheme by comparing with the fourth-order four-stage Runge-Kutta method.

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“…In a similar way, if many steps are performed without output, only three evaluations of ψ [1] and ψ [2] are required per time step.…”

Section: Time Integrationmentioning

confidence: 99%

Section: Time Integrationmentioning

confidence: 99%

Section: Analysis Of the Efficiency Of The Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Time exponential splitting integrator for the Klein–Gordon equation with free parameters in the Hagstrom–Warburton absorbing boundary conditions

Alonso‐Mallo

Portillo

2018

Journal of Computational and Applied Mathematics

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“…The study of the boundedness of the powers (13) is not easy in general (cf. [1]) but, in order to accomplish this, a necessary condition is that the eigenvalues of k(−A)…”

Section: Time Discretization: Exponential Splitting Methodsmentioning

confidence: 99%

“…Therefore, we need 31NM + 98P M + O(N) + O(M) + O(P ) products for every step in time. Moreover, if the last step in the composition of S [2] αk for one step and the first one in S [2] αk for the next step are joined together (ψ [1] αk/2 • ψ [1] αk/2 = ψ [1] αk ), only three times of step 1 are needed, and then the products for one step are…”

Section: Analysis Of the Efficiency Of The Algorithmmentioning

confidence: 99%

Time exponential splitting technique for the Klein–Gordon equation with Hagstrom–Warburton high-order absorbing boundary conditions

Alonso‐Mallo

Portillo

2016

Journal of Computational Physics

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Klein-Gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton (H-W) high-order absorbing boundary conditions (ABCs). Time integration is made by means of exponential splitting schemes that are efficient and easy to implement. In this way, it is possible to achieve a negligible error due to the time integration and to study the behavior of the absorption error. Numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided. The influence of the dispersion coefficient on the error is also studied.

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Numerical resolution of linear evolution multidimensional problems of second order in time

Bujanda

Jorge

Moreta

2010

Numerical Methods Partial

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We present a new class of efficient time integrators for solving linear evolution multidimensional problems of second-order in time named Fractional Step Runge-Kutta-Nyström methods (FSRKN). We show that these methods, combined with suitable spliting of the space differential operator and adequate space discretizations provide important advantages from the computational point of view, mainly parallelization facilities and reduction of computational complexity. In this article, we study in detail the consistency of such methods and we introduce an extension of the concept of R-stability for Runge-Kutta-Nyström methods. We also present some numerical experiments showing the unconditional convergence of a third order method of this class applied to resolve one Initial Boundary Value Problem of second order in time. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 597-620, 2012 Keywords: fractional step Runge-Kutta-Nyström methods I. PRELIMINARIESIn this work, we are concerned with the efficient numerical resolution of second order in time evolution problems governed by partial differential equations which can be written as follows:

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The stability of rational approximations of cosine functions on Hilbert spaces

Cited by 7 publications

References 18 publications

Time exponential splitting integrator for the Klein–Gordon equation with free parameters in the Hagstrom–Warburton absorbing boundary conditions

Time exponential splitting integrator for the Klein–Gordon equation with free parameters in the Hagstrom–Warburton absorbing boundary conditions

Time exponential splitting technique for the Klein–Gordon equation with Hagstrom–Warburton high-order absorbing boundary conditions

Numerical resolution of linear evolution multidimensional problems of second order in time

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