Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb
dc.contributor.author | Boyd, John P. | en_US |
dc.date.accessioned | 2006-09-11T16:03:30Z | |
dc.date.available | 2006-09-11T16:03:30Z | |
dc.date.issued | 2005-03 | en_US |
dc.identifier.citation | Boyd, John P.; (2005). "Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb." Numerical Algorithms 38(1): 197-207. <http://hdl.handle.net/2027.42/45436> | en_US |
dc.identifier.issn | 1572-9265 | en_US |
dc.identifier.issn | 1017-1398 | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/45436 | |
dc.description.abstract | The one-dimensional Helmholtz equation, ε 2 u xx − u = f ( x ), arises in many applications, often as a component of three-dimensional fluids codes. Unfortunately, it is difficult to solve for ε≪1 because the homogeneous solutions are exp (± x /ε), which have boundary layers of thickness O(1/ε). By analyzing the asymptotic Chebyshev coefficients of exponentials, we rederive the Orszag–Israeli rule [16] that Chebyshev polynomials are needed to obtain an accuracy of 1% or better for the homogeneous solutions. (Interestingly, this is identical with the boundary layer rule-of-thumb in [5], which was derived for singular functions like tanh([ x −1]/ε).) Two strategies for small ε are described. The first is the method of multiple scales, which is very general, and applies to variable coefficient differential equations, too. The second, when f ( x ) is a polynomial, is to compute an exact particular integral of the Helmholtz equation as a polynomial of the same degree in the form of a Chebyshev series by solving triangular pentadiagonal systems. This can be combined with the analytic homogeneous solutions to synthesize the general solution. However, the multiple scales method is more efficient than the Chebyshev algorithm when ε is very, very tiny. | en_US |
dc.format.extent | 124933 bytes | |
dc.format.extent | 3115 bytes | |
dc.format.mimetype | application/pdf | |
dc.format.mimetype | text/plain | |
dc.language.iso | en_US | |
dc.publisher | Kluwer Academic Publishers; Springer Science+Business Media | en_US |
dc.subject.other | Theory of Computation | en_US |
dc.subject.other | Mathematics, General | en_US |
dc.subject.other | Algorithms | en_US |
dc.subject.other | Spectral Method | en_US |
dc.subject.other | Algebra | en_US |
dc.subject.other | Chebyshev Polynomials | en_US |
dc.subject.other | Spectral Element Method | en_US |
dc.subject.other | Computer Science | en_US |
dc.subject.other | Numeric Computing | en_US |
dc.title | Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb | en_US |
dc.type | Article | en_US |
dc.subject.hlbsecondlevel | Mathematics | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | Department of Atmospheric, Oceanic and Space Science and Laboratory for Scientific Computation, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI, 48109, USA | en_US |
dc.contributor.affiliationumcampus | Ann Arbor | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/45436/1/11075_2004_Article_2865.pdf | en_US |
dc.identifier.doi | http://dx.doi.org/10.1007/s11075-004-2865-0 | en_US |
dc.identifier.source | Numerical Algorithms | en_US |
dc.owningcollname | Interdisciplinary and Peer-Reviewed |
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