Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime
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.
Fourier pseudospectral spatial approximation
Gautschi-type exponential time integrator
scientific article; zbMATH DE number 6006411
scientific article