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highLY OSCILLATORY DIFFUSION-TYPE EQUATIONS

Journal of Computational Mathematics 2013年第6期31卷 549-572页

作者： Sevda Uskiiplii Altlnbasak Marissa Condon Alfredo Deano Arieh Iserles Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge Cambridge UK School of Electronic Engineering Dublin City University Dublin Ireland Department of Mathematics University Carlos III de Madrid Madrid Spain Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge Cambridge UK

We explore new asymptotic-numeric solvers for partial differential equations with highly oscillatory forcing terms. Such methods represent the solution as an asymptotic series, whose terms can be evaluated by solving non-oscillatory problems and they guarantee high accuracy at a low computational cost. We consider two forms of oscillatory forcing terms, namely when the oscillation is in time or in space： each lends itself to different treatment. Numerical examples highlight the salient features of the new approach.

关键词： Diffusion-type PDEs high oscillation Asymptotic expansions Modulated Fourier expansions.

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On Time-Splitting Pseudospectral Discretization for Nonlinear Klein-Gordon Equation in Nonrelativistic Limit Regime

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Communications in Computational Physics 2014年第7期16卷 440-466页

作者： Xuanchun Dong Zhiguo Xu Xiaofei Zhao Beijing Computational Science Research Center Beijing 100084P.R.China Department of Mathematics National University of SingaporeSingapore 119076Singapore College of Mathematics Jilin UniversityChangchun 130012P.R.China

In this work,we are concerned with a time-splitting Fourier pseudospectral(TSFP)discretization for the Klein-Gordon(KG)equation,involving a dimensionless parameterε∈(0,1].In the nonrelativistic limit regime,the smallεproduces high oscillations in exact solutions with wavelength of O(ε^(−2))in *** key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time,with both the nonlinear and linear subproblems exactly integrable in time and,respectively,Fourier frequency *** method is fully explicit and time ***,we establish rigorously the optimal error bounds of a second-order TSFP for fixedε=O(1),thanks to an observation that the scheme coincides with a type of trigonometric *** the second task,numerical studies are carried out,with special effortsmade to applying the TSFP in the nonrelativistic limit regime,which are geared towards understanding its temporal resolution capacity and meshing strategy for O(ε^(−2))-oscillatory solutions when 0highly oscillatory *** results,either rigorous or numerical,are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.

关键词： Klein-Gordon equation high oscillation time-splitting trigonometric integrator error estimate meshing strategy.

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