Dynamics of Complex Singularities of Nonlinear PDEs

Weideman, J. A. C.

doi:10.1007/978-3-030-86236-7_13

Cited by 6 publications

(1 citation statement)

References 33 publications

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“…Here, we apply exponential asymptotics and transseries to locate singularities and zeros of a solution to a second-order ordinary differential equation (ODE) that comes from studying Burgers' equation in a small-time limit, using the method of transasymptotics developed in [23][24][25]. Our analysis is motivated in part due to significant contemporary interest in the tracking of complex-plane singularities of solutions to differential equations [26][27][28][29][30][31], but also because the methodology presented here should be applicable to other nonlinear ODEs, including those without exact solutions or the kind of special properties that Burgers' equation has (including integrability).…”

Section: Introductionmentioning

confidence: 99%

Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries

Lustri,

Aniceto,

VandenHeuvel

et al. 2023

Proc. R. Soc. A.

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Burgers’ equation is an important mathematical model used to study gas dynamics and traffic flow, among many other applications. Previous analysis of solutions to Burgers’ equation shows an infinite stream of simple poles born at t = 0 + , emerging rapidly from the singularities of the initial condition, that drive the evolution of the solution for t > 0 . We build on this work by applying exponential asymptotics and transseries methodology to an ordinary differential equation that governs the small-time behaviour in order to derive asymptotic descriptions of these poles and associated zeros. Our analysis reveals that subdominant exponentials appear in the solution across Stokes curves; these exponentials become the same size as the leading order terms in the asymptotic expansion along anti-Stokes curves, which is where the poles and zeros are located. In this region of the complex plane, we write a transseries approximation consisting of nested series expansions. By reversing the summation order in a process known as transasymptotic summation, we study the solution as the exponentials grow, and approximate the pole and zero location to any required asymptotic accuracy. We present the asymptotic methods in a systematic fashion that should be applicable to other nonlinear differential equations.

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Section: Introductionmentioning

confidence: 99%

Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries

Lustri,

Aniceto,

VandenHeuvel

et al. 2023

Proc. R. Soc. A.

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Numerical analytic continuation

Trefethen

2023

Japan J. Indust. Appl. Math.

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Let f be an analytic function on a simply-connected compact continuum E of the complex z-plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points $$z\in {\mathbb C}$$ z ∈ C ? How can we estimate the locations of real or complex singularities of f? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations of f by the AAA algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs.

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Infinite-order accuracy limit of finite difference formulas in the complex plane

Fornberg

2022

IMA Journal of Numerical Analysis

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It was recently found that finite difference (FD) formulas become remarkably accurate when approximating derivatives of analytic functions $f(z)$ in the complex $z=x+\text{i}y$ plane. On unit-spaced grids in the $x,y$-plane, the FD weights decrease to zero with the distance to the stencil center at a rate similar to that of a Gaussian, typically falling below the level of double precision accuracy $\mathcal{O}(10^{-16})$ already about four node spacings away from the center point. We follow up on these observations here by analyzing and illustrating the features of such FD stencils in their infinite-order accurate limit (for traditional FD approximations known as their pseudospectral limit).

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Dynamics of Complex Singularities of Nonlinear PDEs

Cited by 6 publications

References 33 publications

Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries

Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries

Numerical analytic continuation

Infinite-order accuracy limit of finite difference formulas in the complex plane

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