An improved Talbot method for numerical Laplace transform inversion
The classical Talbot method for the computation of the inverse Laplace transform is improved for the case where the transform is analytic in the complex plane except for the negative real axis. First, by using a truncated Talbot contour rather than the classical contour that goes to infinity in the...
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| Published in | Numerical algorithms Vol. 68; no. 1; pp. 167 - 183 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.01.2015
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 |
| DOI | 10.1007/s11075-014-9895-z |
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| Abstract | The classical Talbot method for the computation of the inverse Laplace transform is improved for the case where the transform is analytic in the complex plane except for the negative real axis. First, by using a truncated Talbot contour rather than the classical contour that goes to infinity in the left half-plane, faster convergence is achieved. Second, a control mechanism for improving numerical stability is introduced. |
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| AbstractList | The classical Talbot method for the computation of the inverse Laplace transform is improved for the case where the transform is analytic in the complex plane except for the negative real axis. First, by using a truncated Talbot contour rather than the classical contour that goes to infinity in the left half-plane, faster convergence is achieved. Second, a control mechanism for improving numerical stability is introduced. |
| Author | Dingfelder, Benedict Weideman, J. A. C. |
| Author_xml | – sequence: 1 givenname: Benedict surname: Dingfelder fullname: Dingfelder, Benedict organization: Zentrum Mathematik, M3, Technische Universität München – sequence: 2 givenname: J. A. C. surname: Weideman fullname: Weideman, J. A. C. email: weideman@sun.ac.za organization: Department of Mathematical Sciences, Stellenbosch University |
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| Cites_doi | 10.1137/050625837 10.1016/j.apnum.2004.06.015 10.1002/nme.995 10.1145/78928.78932 10.1007/s10543-006-0077-9 10.1145/212066.212068 10.1093/imanum/drn074 10.1145/155743.155788 10.1093/imanum/23.2.269 10.1090/S0025-5718-07-01945-X 10.1007/978-1-4612-0599-9 10.1137/050629653 10.1137/040611045 10.1093/imamat/23.1.97 |
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| References_xml | – reference: KressRNumerical analysis1998New YorkSpringer-Verlag10.1007/978-1-4612-0599-90913.65001 – reference: Abate, J., Valkó, P.P.: NumericalLaplaceInversion (2002). http://library.wolfram.com/infocenter/MathSource/4738/ – reference: WeidemanJACOptimizing Talbot’s contours for the inversion of the Laplace transformSIAM J. Numer. Anal20064462342236210.1137/0506258371131.651052272597 – reference: SheenDSloanIHThoméeVA parallel method for time discretization of parabolic equations based on Laplace transformation and quadratureIMA J. Numer. Anal.200323226929910.1093/imanum/23.2.2691022.651081975267 – reference: Butcher, J.C.: On the numerical inversion of Laplace and Mellin transforms. Conference on Data Processing and Automatic Computing Machines,Salisbury,Australia (1957) – reference: López-FernándezMPalenciaCOn the numerical inversion of the Laplace transform of certain holomorphic mappingsAppl. Numer. Math.2004512-328930310.1016/j.apnum.2004.06.0151059.651202091405 – reference: DuffyDGOn the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applicationsACM Trans. Math. Software199319333335910.1145/155743.1557880892.650791340860 – reference: López-FernándezMPalenciaCSchädleAA spectral order method for inverting sectorial Laplace transformsSIAM J. Numer. Anal20064431332135010.1137/0506296531124.651202231867 – reference: GavrilyukIPMakarovVLExponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spacesSIAM J. Numer. Ana.l20054352144217110.1137/0406110451116.650632192335 – reference: Dingfelder, B.: Raten- und stabilitätsoptimierte Algorithmen zur Berechnung eines Integrals mit Hankelintegrationsweg. Bachelor’s thesis, Technische Universität München,Germany (2012) – reference: AbateJValkóPPMulti-precision laplace transform inversionInt. J. Numer. Methods Fluids200460597999310.1002/nme.9951059.65118 – reference: RizzardiMA modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace transformACM Trans. Math. Software199521434737110.1145/212066.2120680887.651331364695 – reference: StrangGComputational science and engineering2007WellesleyWellesley-Cambridge Press1194.65001 – reference: TrefethenLNWeidemanJACSchmelzerTTalbot quadratures and rational approximationsBIT200646365367010.1007/s10543-006-0077-91103.650302265580 – reference: McClure, T. talbot_inversion (2013). http://www.mathworks.com/matlabcentral/fileexchange/39035-numerical-inverse-laplace-transform/ – reference: Dingfelder, B., Weideman, J.A.C.: ModifiedTalbot (2014). http://arxiv.org/e-print/1304.2505v3 – reference: MurliARizzardiMAlgorithm 682: Talbot’s method for the Laplace inversion problemACM Trans. Math. Software199016215816810.1145/78928.789320900.65374 – reference: WeidemanJACImproved contour integral methods for parabolic PDEsIMA J. Numer. Anal.201030133435010.1093/imanum/drn0741186.651252580562 – reference: WeidemanJACTrefethenLNParabolic and hyperbolic contours for computing the Bromwich integralMath. Comp.2007762591341135610.1090/S0025-5718-07-01945-X1113.651192299777 – reference: TalbotAThe accurate numerical inversion of Laplace transformsJ. Inst. Math. Appl19792319712010.1093/imamat/23.1.970406.65054526286 – reference: Trefethen, L.N.: Private communication (2006) – volume: 44 start-page: 2342 issue: 6 year: 2006 ident: 9895_CR19 publication-title: SIAM J. Numer. Anal doi: 10.1137/050625837 – volume: 51 start-page: 289 issue: 2-3 year: 2004 ident: 9895_CR9 publication-title: Appl. Numer. Math. doi: 10.1016/j.apnum.2004.06.015 – volume: 60 start-page: 979 issue: 5 year: 2004 ident: 9895_CR2 publication-title: Int. J. Numer. Methods Fluids doi: 10.1002/nme.995 – volume: 16 start-page: 158 issue: 2 year: 1990 ident: 9895_CR12 publication-title: ACM Trans. Math. Software doi: 10.1145/78928.78932 – ident: 9895_CR11 – volume: 46 start-page: 653 issue: 3 year: 2006 ident: 9895_CR18 publication-title: BIT doi: 10.1007/s10543-006-0077-9 – volume: 21 start-page: 347 issue: 4 year: 1995 ident: 9895_CR13 publication-title: ACM Trans. Math. Software doi: 10.1145/212066.212068 – volume: 30 start-page: 334 issue: 1 year: 2010 ident: 9895_CR20 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/drn074 – volume: 19 start-page: 333 issue: 3 year: 1993 ident: 9895_CR6 publication-title: ACM Trans. Math. Software doi: 10.1145/155743.155788 – ident: 9895_CR5 – volume-title: Computational science and engineering year: 2007 ident: 9895_CR15 – volume: 23 start-page: 269 issue: 2 year: 2003 ident: 9895_CR14 publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/23.2.269 – ident: 9895_CR3 – volume: 76 start-page: 1341 issue: 259 year: 2007 ident: 9895_CR21 publication-title: Math. Comp. doi: 10.1090/S0025-5718-07-01945-X – ident: 9895_CR1 – ident: 9895_CR4 – volume-title: Numerical analysis year: 1998 ident: 9895_CR8 doi: 10.1007/978-1-4612-0599-9 – ident: 9895_CR17 – volume: 44 start-page: 1332 issue: 3 year: 2006 ident: 9895_CR10 publication-title: SIAM J. Numer. Anal doi: 10.1137/050629653 – volume: 43 start-page: 2144 issue: 5 year: 2005 ident: 9895_CR7 publication-title: SIAM J. Numer. Ana.l doi: 10.1137/040611045 – volume: 23 start-page: 97 issue: 1 year: 1979 ident: 9895_CR16 publication-title: J. Inst. Math. Appl doi: 10.1093/imamat/23.1.97 |
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| StartPage | 167 |
| SubjectTerms | Algebra Algorithms Computer Science Contours Convergence Half planes Infinity Inverse Laplace transforms Mathematical analysis Mathematical models Numeric Computing Numerical Analysis Numerical stability Original Paper Shape Theory of Computation Transforms |
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| Title | An improved Talbot method for numerical Laplace transform inversion |
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