Uniform Error Bounds of an Exponential Wave Integrator for the Long-Time Dynamics of the Nonlinear Klein--Gordon Equation
نویسندگان
چکیده
We establish uniform error bounds of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics nonlinear Klein--Gordon equation (NKGE) with a cubic nonlinearity whose strength is characterized by $\varepsilon^2$ $\varepsilon \in (0, 1]$ dimensionless parameter. When $0 < \varepsilon \ll 1$, problem equivalent to NKGE small initial data (and $O(1)$ nonlinearity), while amplitude solution) at $O(\varepsilon)$. For up time $O(1/\varepsilon^{2})$, resolution and classical numerical methods depend significantly on parameter $\varepsilon$, which causes severe burdens as \to 0^+$. The EWI-FP fully explicit, symmetric in time, has many superior properties solving equations. By adapting energy combined mathematical induction, we rigorously carry out discretization $O(h^{m_0} + \varepsilon^{2-\beta}\tau^2)$ $O(1/\varepsilon^{\beta})$ \leq \beta 2$, mesh size $h$, step $\tau$ $m_0$ integer depending regularity solution. rescaling our results are straightforwardly extended those $\varepsilon$-scalability (or meshing strategy requirement) oscillatory NKGE, solution propagates waves wavelength $O(\varepsilon^{\beta})$ space respectively, speed $O(\varepsilon^{-\beta})$. Finally, extensive reported confirm estimates.
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ژورنال
عنوان ژورنال: Multiscale Modeling & Simulation
سال: 2021
ISSN: ['1540-3459', '1540-3467']
DOI: https://doi.org/10.1137/20m1327677