Improved Uniform Error Bounds on Time-Splitting Methods for Long-Time Dynamics of the Nonlinear Klein--Gordon Equation with Weak Nonlinearity
نویسندگان
چکیده
We establish improved uniform error bounds on a second-order Strang time-splitting method which is equivalent to an exponential wave integrator for the long-time dynamics of nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength characterized by $\varepsilon^2$ $0 < \varepsilon \leq 1$ dimensionless parameter. Actually, when \ll 1$, NKGE $O(\varepsilon^2)$ nonlinearity and $O(1)$ initial data that small data, amplitude at $O(\varepsilon)$. begin semidiscretization derive full-discretization Fourier spectral in space. Employing regularity compensation oscillation technique controls high frequency modes exact solution analyzes low phase cancellation energy method, we carry out $O(\varepsilon^2\tau^2)$ $O(h^m+\varepsilon^2\tau^2)$ up long time $T_\varepsilon = T/\varepsilon^2$ $T$ fixed, respectively. Extensions higher-order methods case oscillatory complex are also discussed. Finally, numerical results provided confirm demonstrate they sharp.
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ژورنال
عنوان ژورنال: SIAM Journal on Numerical Analysis
سال: 2022
ISSN: ['0036-1429', '1095-7170']
DOI: https://doi.org/10.1137/21m1449774