Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of on are proportional to where Λ( j ) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f ( x )= x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergencePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43417/1/10915_2005_Article_9010.pd

Parallel Spectral Fourier Algorithm for Fluid Dynamics Problems

We present a high-order parallel algorithm which requires only the minimum inter-processor communication dictated by the physical nature of the problem at hand. This algorithm is applied to the incompressible Navier-Stokes equations. The parallelization is achieved by domain decomposition. A novel feature of the present approach is that the spatial discretization in subdomains is performed using the Fourier method. To avoid the Gibbs phenomenon, the global functions are decomposed into smooth local pieces. Then the Fourier method is applied on extended local subdomains with spectral accuracy. The continuity conditions on the interfaces are enforced by adding the homogeneous solutions. Such solutions often have fast decay properties which can be utilized to minimize interprocessor communication. In effect, an overwhelming part of the computation is performed independently in subdomains (processors) or using only local communication. The present method allows the treatment of problems ..

Spectral Multidomain Technique with Local Fourier Basis

A novel domain decomposition method for spectrally accurate solutions of PDEs is presented. A Local Fourier Basis technique is adapted for the construction of the elemental solutions in subdomains. C 1 continuity is achieved on the interfaces by a matching procedure using the analytical homogeneous solutions of a one dimensional equation. The method can be applied to the solution of elliptic problems of the Poisson or Helmholtz type as well as to time discretized parabolic problems in one or more dimensions. The accuracy is tested for several stiff problems with steep solutions. The present domain decomposition approach is particularly suitable for parallel implementations, in particular, on MIMD type parallel machines. 1 Introduction The numerical solution of multidimensional non-linear evolution equations (and especially the particularly difficult case of the incompressible Navier- Stokes equations) is a heavy computational task. At the range of parameters, which is of interest in..

Multidomain Local Fourier Method for PDEs in Complex Geometries

A low communication parallel algorithm is developed for the solution of timedependent non-linear PDEs. The parallelization is achieved by domain decomposition. The discretization in time is performed via a third order semi-implicit stiffly stable scheme. The elemental solutions in the subdomains are constructed using a high-order method with the Local Fourier Basis (LFB). The continuity of the global solution is accomplished by a point-wise matching of the local subsolutions on the interfaces. The matching relations are derived in terms of the jumps on the interfaces. The LFB method enables splitting a 2-D problem with a global coupling of the interface unknowns into a set of uncoupled 1-D differential equations. Localization properties of an elliptic operator, resulting from the discretization in time of a time-dependent problem, are utilized in order to simplify the matching relations. In effect, only local (neighbor-to-neighbor) communication between the processors becomes necessary..

Domain Decomposition Methods for Solving Parabolic PDEs on Multiprocessors

Time-dependent partial differential equations of parabolic type describe processors for which the influence of remote regions is practically negligible. This property of locality can be exploit in domain decomposition techniques to simplify the global matching relations among subdomains. This paper presents two spectral multidomain approaches for the solution of parabolic PDE's when only minimal communication between adjacent subdomains is required. A breaf survey of some principle domain decomposition approaches for solving elliptic and parabolic problems is also provided. 1 Introduction In the past several years computational algorithms based on domain decomposition (DD) techniques have attracted much attention because of their ability to facilitate the solution of very large problems on machines with limited storage. They are also often used to design adaptive algorithms to capture steep gradients that appear in the solution of certain problems [26] or to enable the coupling of reg..