Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems

Okayama, Tomoaki

doi:10.1007/s10543-017-0663-z

Cited by 8 publications

(10 citation statements)

References 17 publications

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“…holds for all z ∈ ψ(D d ). Let n be a positive integer, and let h be selected by (10). Then, there exists a constant C independent of n such that…”

Section: Convergence Theorems For (Se1) (Se2) and (Se3)mentioning

confidence: 99%

“…Let µ = min{α, β}, let n be a positive integer, and let h be selected by (10). Furthermore, let M and N be positive integers defined by (11).…”

Section: Convergence Theorems For (Se1) (Se2) and (Se3)mentioning

confidence: 99%

“…Assume that f is analytic in φ(D d ) for d with 0 < d < π/2, and that there exist positive constants K and µ with µ ≤ 1 such that (12) holds for all z ∈ φ(D d ). Let n be a positive integer, and let h be selected by (10). Then, there exists a constant C independent of n such that…”

Section: Convergence Theorems For (De1) (De2) and (De3)mentioning

confidence: 99%

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Yet another DE-Sinc indefinite integration formula

Okayama¹,

Tanaka²

2022

Preprint

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Based on the Sinc approximation combined with the tanh transformation, Haber derived an approximation formula for numerical indefinite integration over the finite interval (-1, 1). The formula, (SE1) uses a special function for the basis functions. In contrast, Stenger derived another formula (SE2), which does not use any special function but does include a double sum. Subsequently, Muhammad and Mori proposed the formula (DE1), which replaces the tanh transformation with the double-exponential transformation in the formula (SE1). Almost simultaneously, Tanaka et al. proposed the formula (DE2), which was based on the same replacement in (SE2). As they reported, the replacement drastically improves the convergence rate of (SE1) and (SE2). In addition to the formulas above, Stenger derived yet another indefinite integration formula (SE3) based on the Sinc approximation combined with the tanh transformation, which has an elegant matrix-vector form. In this paper, we propose the replacement of the tanh transformation with the double-exponential transformation in the formula (SE3). We provide a theoretical analysis as well as a numerical comparison.

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“…holds for all z ∈ ψ(D d ). Let n be a positive integer, and let h be selected by (10). Then, there exists a constant C independent of n such that…”

Section: Convergence Theorems For (Se1) (Se2) and (Se3)mentioning

confidence: 99%

“…Let µ = min{α, β}, let n be a positive integer, and let h be selected by (10). Furthermore, let M and N be positive integers defined by (11).…”

Section: Convergence Theorems For (Se1) (Se2) and (Se3)mentioning

confidence: 99%

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Yet another DE-Sinc indefinite integration formula

Okayama¹,

Tanaka²

2022

Preprint

Self Cite

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show abstract

“…Under certain assumptions on K(t) and g(t), it was shown in [8,19,20] that the linear Sinc system (7) with a sufficiently large N is uniquely solvable, and the obtained Sinc approximation y h (t) in ( 2) determined by the solution vector y h converges exponentially, that is…”

Section: The Constant Coefficient Casementioning

confidence: 99%

Parallel-in-time preconditioners for the Sinc-Nyström method

Li¹,

Wu²

2021

Preprint

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The Sinc-Nyström method in time is a high-order spectral method for solving evolutionary differential equations and it has wide applications in scientific computation. But in this method we have to solve all the time steps implicitly at one-shot, which may results in a large-scale nonsymmetric dense system that is expensive to solve. In this paper, we propose and analyze a parallel-in-time (PinT) preconditioner for solving such Sinc-Nyström systems, where both the parabolic and hyperbolic PDEs are investigated. Attributed to the special Toeplitz-like structure of the Sinc-Nyström systems, the proposed PinT preconditioner is indeed a low-rank perturbation of the system matrix and we show that the spectrum of the preconditioned system is highly clustered around one, especially when the time step size is refined. Such a clustered spectrum distribution matches very well with the numerically observed mesh-independent GMRES convergence rates in various examples. Several linear and nonlinear ODE and PDE examples are presented to illustrate the convergence performance of our proposed PinT preconditioners, where the achieved exponential order of accuracy are especially attractive to those applications in need of high accuracy.

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“…Sinc methods have been proposed and studied by Stenger [21]. The sinc method has been increasingly used for solving ordinary differential equations [22][23][24][25], partial differential equations [26][27][28][29], and integral equations [30][31][32][33]; it not only has exponential convergence rate, but also can deal with singular problems well. In recent years, the sinc method has been also frequently employed for the numerical solution of fractional partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

Chen

2020

Adv Differ Equ

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In this paper, a new numerical algorithm for solving the time fractional convection–diffusion equation with variable coefficients is proposed. The time fractional derivative is estimated using the $L_{1}$ L 1 formula, and the spatial derivative is discretized by the sinc-Galerkin method. The convergence analysis of this method is investigated in detail. The numerical solution is $2-\alpha$ 2 − α order accuracy in time and exponential rate of convergence in space. Finally, some numerical examples are given to show the effectiveness of the numerical scheme.

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Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems

Cited by 8 publications

References 17 publications

Yet another DE-Sinc indefinite integration formula

Yet another DE-Sinc indefinite integration formula

Parallel-in-time preconditioners for the Sinc-Nyström method

Sinc-Galerkin method for solving the time fractional convection–diffusion equation with variable coefficients

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