Deciphering singularities by discrete methods

Tourigny, Yves; Grinfeld, Michael

doi:10.1090/s0025-5718-1994-1203737-5

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Introduction Many Nonlinear Pdes Have Solutions That Become1

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2022

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Cited by 13 publications

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“…One example is the study reported in [14], in which Padé approximation was used to study singularity formation in a fluid flow problem. Another study of this type is [17], which includes a test problem similar to (1.2).…”

Section: Introduction Many Nonlinear Pdes Have Solutions That Becomementioning

confidence: 99%

Computing the Dynamics of Complex Singularities of Nonlinear PDEs

Weideman¹

2003

SIAM J. Appl. Dyn. Syst.

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Abstract.A two-step strategy is proposed for the computation of singularities in nonlinear PDEs. The first step is the numerical solution of the PDE using a Fourier spectral method; the second step involves numerical analytical continuation into the complex plane using the epsilon algorithm to sum the Fourier series. Test examples include the inviscid Burgers and nonlinear heat equations as well as a transport equation involving the Hilbert transform. Numerical results, including Web animations that show the dynamics of the singularities in the complex plane, are presented.

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Section: Introduction Many Nonlinear Pdes Have Solutions That Becomementioning

confidence: 99%

Computing the Dynamics of Complex Singularities of Nonlinear PDEs

Weideman¹

2003

SIAM J. Appl. Dyn. Syst.

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Dynamics of Complex Singularities of Nonlinear PDEs

Weideman

2022

SEMA SIMAI Springer Series

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Solutions to nonlinear evolution equations exhibit a wide range of interesting phenomena such as shocks, solitons, recurrence, and blow-up. As an aid to understanding some of these features, the solutions can be viewed as analytic functions of a complex space variable. The dynamics of poles and branch point singularities in the complex plane can often be associated with the aforementioned features of the solution. Some of the computational and analytical results in this area are surveyed here. This includes a first attempt at computing the poles in the famous Zabusky–Kruskal experiment that lead to the discovery of the soliton.

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Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations

Majidian¹

2018

Bit Numer Math

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Deciphering singularities by discrete methods

Cited by 13 publications

References 18 publications

Computing the Dynamics of Complex Singularities of Nonlinear PDEs

Computing the Dynamics of Complex Singularities of Nonlinear PDEs

Dynamics of Complex Singularities of Nonlinear PDEs

Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations

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