Journal ArticleDOI
Computing the Dynamics of Complex Singularities of Nonlinear PDEs
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A two-step strategy is proposed for the computation of singularities in nonlinear PDEs using a Fourier spectral method and the epsilon algorithm to sum the Fourier series.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.read more
Citations
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A Rational Spectral Collocation Method with Adaptively Transformed Chebyshev Grid Points
T. W. Tee,Lloyd N. Trefethen +1 more
TL;DR: Numerical experiments indicate that the method far outperforms the standard Chebyshev spectral collocation method for problems whose solutions have singularities in the complex plane close to $[-1,1]$.
Journal ArticleDOI
Applied and computational complex analysis. Vol. 1. Power series, integration, conformal mapping, location of zeros: Henrici P. xv + 682 pp., John Wiley and Sons, Inc., New York — London, 1974☆
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Numerical study of shock formation in the dispersionless Kadomtsev-Petviashvili equation and dispersive regularizations
Christian Klein,Kristelle Roidot +1 more
TL;DR: In this article, the formation of singularities in solutions to the dispersionless Kadomtsev-Petviashvili (dKP) equation is studied numerically for different classes of initial data.
Journal ArticleDOI
Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms
TL;DR: In this paper, the authors show that if f(x) is analytic on the expansion interval, then all the coefficients of the inverse power series are zero and the problem becomes one of "beyond-all-orders" or "exponential" asymptotics.
Journal ArticleDOI
Analysis of complex singularities in high-Reynolds-number Navier-Stokes solutions
TL;DR: In this article, numerical solutions of the laminar Prandtl boundary-layer and Navier-Stokes equations are considered for the case of the two-dimensional uniform flow past an impulsively-started circular cylinder.
References
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Journal ArticleDOI
On the Lambert W function
TL;DR: A new discussion of the complex branches of W, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W are presented.
Book
Spectral Methods in Fluid Dynamics
M. Y. Hussaini,Thomas A. Zang +1 more
TL;DR: Spectral methods have been widely used in simulation of stability, transition, and turbulence as discussed by the authors, and their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed.
Journal ArticleDOI
Spectral Methods in Fluid Dynamics.
TL;DR: In this article, the authors present a set of methods for the estimation of two-dimensional fluid flow, including a Fourier Galerkin method and a Chebyshev Collocation method.