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Register a new userTheoretical analysis of Sinc-collocation methods and Sinc-Nystr{o}m methods for initial value problems
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Tomoaki Okayama
Publication date
2013
fields
Informatics Engineering
and research's language is
English
Authors
Tomoaki Okayama
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A Sinc-collocation method has been proposed by Stenger, and he also gave theoretical analysis of the method in the case of a `scalar equation. This paper extends the theoretical results to the case of a `system of equations. Furthermore, this paper proposes more efficient method by replacing the variable transformation employed in Stengers method. The efficiency is confirmed by both of theoretical analysis and numerical experiments. In addition to the existing and newly-proposed Sinc-collocation methods, this paper also gives similar theoretical results for Sinc-Nystr{o}m methods proposed by Nurmuhammad et al. From a viewpoint of the computational cost, it turns out that the newly-proposed Sinc-collocation method is the most efficient among those methods.
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The Sinc-Nystr{o}m 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-Nystr{o}m systems, where both the parabolic and hyperbolic PDEs are investigated. Attributed to the special Toeplitz-like structure of the Sinc-Nystr{o}m 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.
This work focuses on the construction of a new class of fourth-order accurate methods for multirate time evolution of systems of ordinary differential equations. We base our work on the Recursive Flux Splitting Multirate (RFSMR) version of the Multirate Infinitesimal Step (MIS) methods and use recent theoretical developments for Generalized Additive Runge-Kutta methods to propose our higher-order Relaxed Multirate Infinitesimal Step extensions. The resulting framework supports a range of attractive properties for multirate methods, including telescopic extensions, subcycling, embeddings for temporal error estimation, and support for changes to the fast/slow time-scale separation between steps, without requiring any sacrifices in linear stability. In addition to providing rigorous theoretical developments for these new methods, we provide numerical tests demonstrating convergence and efficiency on a suite of multirate test problems.
This paper reinforces numerical iterated integration developed by Muhammad--Mori in the following two points: 1) the approximation formula is modified so that it can achieve a better convergence rate in more general cases, and 2) explicit error bound is given in a computable form for the modified formula. The formula works quite efficiently, especially if the integrand is of a product type. Numerical examples that confirm it are also presented.
A Sinc-Nystrom method for Volterra integro-differential equations was developed by Zarebnia in 2010. The method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity. However, its exponential convergence has not been proved theoretically. In addition, to implement the method, the regularity of the solution is required, although the solution is an unknown function in practice. This paper reinforces the method by presenting two theoretical results: 1) the regularity of the solution is analyzed, and 2) its convergence rate is rigorously analyzed. Moreover, this paper improves the method so that a much higher convergence rate can be attained, and theoretical results similar to those listed above are provided. Numerical comparisons are also provided.
The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation. Their convergence rates have been analyzed for typical cases including finite, semi-infinite, and infinite intervals. In addition, for verified automatic integration, more explicit error bounds that are computable have been recently given on a finite interval. In this paper, such explicit error bounds are given in the remaining cases on semi-infinite and infinite intervals.
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