Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime
type: Item
instance of: scholarly article
title: Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime
zbMATH Open document ID: 1248.65087
author: Weizhu Bao
author: Xuanchun Dong
published in: Numerische Mathematik
publication date: 2012-02-14T00:00:00Z
review text: The paper deals with the dimensionsless relativistic Klein-Gordon (KG) equation in 1, 2 and 3 spatial dimensions \[ \varepsilon^2 \partial_{tt}u-\Delta u+\frac{1}{\varepsilon^2}u+f(u)=0, \quad {\mathbf {x}} \in \mathbb{R}^{d}, \] with the initial conditions \[ u({\mathbf {x}},0)=\phi({\mathbf {x}}), ~\partial_{t}u({\mathbf {x}},0)=\frac{1}{\varepsilon^2} \gamma({\mathbf {x}}), \] where \(\varepsilon\) is a small parameter, \(\phi, \gamma\) are given real-valued functions, \(f(u)\) is a real-valued function independent of \(\varepsilon\) and satisfies \(f(0)=0\). The authors analyse the four frequently used finite difference time domain (FDTM) methods and pay particulary attention to how the error bounds depend explicitely on the mesh size \(h\) and the time step \(\tau\) as well as the small parameter \(\varepsilon\). Based on the estimates, it is shown that the meshing strategy requirement (\(\varepsilon\)-scalability) for those frequently used FDTD methods should be \(\tau={\mathcal {O}}(\varepsilon^3), h={\mathcal {O}}(1)\) to provide ``correct'' numerical results which suggests high computational costs for small \(\varepsilon\). The authors propose a method based on the Fourier pseudospectral or centered finite difference spatial discretization and the Gautschi-type exponential time integrator. It is shown that the \(\varepsilon\)-scalability is improved to \(\tau={\mathcal {O}}(\varepsilon^2), h={\mathcal {O}}(1)\) for the nonlinear and to \(\tau={\mathcal {O}}(1), ~h={\mathcal {O}}(1)\) for the linear KG equation.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 35Q75
Mathematics Subject Classification ID: 65M15
zbMATH DE Number: 6006411
zbMATH Keywords: small parameter
zbMATH Keywords: finite difference time domain methods
zbMATH Keywords: Fourier pseudospectral spatial approximation
zbMATH Keywords: finite difference spatial approximation
zbMATH Keywords: relativistic Klein-Gordon equation
zbMATH Keywords: error bounds
zbMATH Keywords: \(\varepsilon\)-scalability
zbMATH Keywords: numerical results
zbMATH Keywords: Gautschi-type exponential time integrator
reviewed by: Iwan Gawriljuk
MaRDI profile type: Publication
full work available at URL: s00211-011-0411-2
OpenAlex ID: W2064726339
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DOI: 10.1007/S00211-011-0411-2
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