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Register a new userExplicit error bound for modified numerical iterated integration by means of Sinc methods
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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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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.
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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.
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.
Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing restrictions on timestep size. However, implicit methods require solutions to one or more systems of nonlinear equations at each timestep, which for large simulations can be prohibitively expensive. This paper introduces a new family of linearly implicit multistep methods (LIMM), which only requires the solution of one linear system per timestep. Order conditions and stability theory for these methods are presented, as well as design and implementation considerations. Practical methods of order up to five are developed that have similar error coefficients, but improved stability regions, when compared to the widely used BDF methods. Numerical testing of a self-starting variable stepsize and variable order implementation of the new LIMM methods shows measurable performance improvement over a similar BDF implementation.
This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebraic equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study the numerical homogenization for multi-scale elliptic PDEs with monotone nonlinearity, in particular the Leray-Lions problem (a prototypical example is the p-Laplacian equation), where the nonlinearity cannot be parameterized with low dimensional parameters, and the linearization error is non-negligible. We develop the iterated numerical homogenization scheme by combining numerical homogenization methods for linear equations, and the so-called quasi-norm based iterative approach for monotone nonlinear equation. We propose a residual regularized nonlinear iterative method, and in addition, develop the sparse updating method for the efficient update of coarse spaces. A number of numerical results are presented to complement the analysis and valid the numerical method.
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