Multiple-Scale Solution of Initial-Boundary Value Problems for Weakly Nonlinear Wave Equations on the Semi-Infinite Line

Chikwendu, S. C.; Easwaran, C. V.

doi:10.1137/0152054

Cited by 9 publications

(4 citation statements)

References 11 publications

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“…The 'exact' solution is obtained numerically by the exponential-wave integrator Fourier pseudospectral method [5,19] with a very fine mesh size and a very small time step, e.g. h e = π/2 15 and τ e = 10 −5 . Denote u n h,τ as the numerical solution at time t = t n obtained by a numerical method with mesh size h and time step τ.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…There are extensive analytical results in the literature for the NKGE (1.1) (or (1.3)). For the existence of global classical solutions and almost periodic solutions as well as asymptotic behavior of solutions, we refer to [10][11][12]15,[40][41][42] and references therein. For the Cauchy problem with small initial data (or weak nonlinearity), the global existence and asymptotic behavior of solutions were studied in different space dimensions and with different nonlinear terms [25,26,31,35,38].…”

Section: Introductionmentioning

confidence: 99%

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Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Bao¹

2019

CICP

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We establish error bounds of the finite difference time domain (FDTD) methods for the long time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity, while the nonlinearity strength is characterized by ε 2 with 0<ε≤1 a dimensionless parameter. When 0 < ε ≪ 1, it is in the weak nonlinearity regime and the problem is equivalent to the NKGE with small initial data, while the amplitude of the initial data (and the solution) is at O(ε). Four different FDTD methods are adapted to discretize the problem and rigorous error bounds of the FDTD methods are established for the long time dynamics, i.e. error bounds are valid up to the time at O(1/ε β ) with 0 ≤ β ≤ 2, by using the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. In the error bounds, we pay particular attention to how error bounds depend explicitly on the mesh size h and time step τ as well as the small parameter ε ∈ (0,1], especially in the weak nonlinearity regime when 0 < ε ≪ 1. Our error bounds indicate that, in order to get "correct" numerical solutions up to the time at O(1/ε β ), the ε-scalability (or meshing strategy) of the FDTD methods should be taken as: h =O(ε β/2 ) and τ =O(ε β/2 ). As a by-product, our results can indicate error bounds and ε-scalability of the FDTD methods for the discretization of an oscillatory NKGE which is obtained from the case of weak nonlinearity by a rescaling in time, while its solution propagates waves with wavelength at O(1) in space and O(ε β ) in time. Extensive numerical results are reported to confirm our error bounds and to demonstrate that they are sharp.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Bao¹

2019

CICP

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“…In [3], Chikwendu and Easwaran show that this particular combination of temporal and spatial scales is appropriate for weakly nonlinear wave equations of the form where 0 < ε≪ 1. Our work generalizes these findings to encompass pairs of hyperbolic conservation laws that arise naturally in many physical examples.…”

Section: Introductionmentioning

confidence: 99%

Initial Boundary‐Value Problems for a Pair of Conservation Laws

Yong

Kevorkian

2002

Stud Appl Math

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We describe a multiple-scale technique for solving the initial boundary-value problem over the positive x-axis for a one-dimensional pair of hyperbolic conservation laws. This technique involves decomposing the solution into waves and incorporating slow temporal and stretched spatial scales in different parts of the solution domain. We apply these ideas to a wavemaker problem for shallow water flow and show why the presence of source terms in the conservation laws makes the analytic solution more complicated.

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“…Introduction. In this paper, we consider the following nonlinear Klein-Gordon equation (NKGE) [12,28,29,31,42] (1.1) ∂ tt u(x, t) − ∆u(x, t) + u(x, t) + ε 2 u 3 (x, t) = 0, x ∈ Ω, t > 0, u(x, 0) = u 0 (x), ∂ t u(x, 0) = u 1 (x), x ∈ Ω.…”

mentioning

confidence: 99%

Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein--Gordon equation with weak nonlinearity

Bao¹,

Cai²,

Feng³

2021

Preprint

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We establish improved uniform error bounds on time-splitting methods for the longtime dynamics of the nonlinear Klein-Gordon equation (NKGE) with weak cubic nonlinearity, whose strength is characterized by ε 2 with 0 < ε ≤ 1 a dimensionless parameter. Actually, when 0 < ε ≪ 1, the NKGE with O(ε 2 ) nonlinearity and O(1) initial data is equivalent to that with O(1) nonlinearity and small initial data of which the amplitude is at O(ε). We begin with a semi-discretization of the NKGE by the second-order time-splitting method, and followed by a full-discretization via the Fourier spectral method in space. Employing the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the exact solution and analyzes the low frequency modes by phase cancellation and energy method, we carry out the improved uniform error bounds at O(ε 2 τ 2 ) and O(h m + ε 2 τ 2 ) for the second-order semi-discretization and full-discretization up to the long time Tε = T /ε 2 with T fixed, respectively. Extensions to higher order time-splitting methods and the case of an oscillatory complex NKGE are also discussed. Finally, numerical results are provided to confirm the improved error bounds and to demonstrate that they are sharp.

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Multiple-Scale Solution of Initial-Boundary Value Problems for Weakly Nonlinear Wave Equations on the Semi-Infinite Line

Cited by 9 publications

References 11 publications

Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Long Time Error Analysis of Finite Difference Time Domain Methods for the Nonlinear Klein-Gordon Equation with Weak Nonlinearity

Initial Boundary‐Value Problems for a Pair of Conservation Laws

Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein--Gordon equation with weak nonlinearity

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