Uniform Error Bounds of an Exponential Wave Integrator for the Long-Time Dynamics of the Nonlinear Klein--Gordon Equation

Abstract

We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is O(ε) with 0 < ε ≪ 1 a dimensionless parameter up to the time at O(1/ε 2 ). The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discreization at O(ε 2 τ ) instead of O(τ ) according to classical error estimates and at O(h m + ε 2 τ ) for the full-discretization up to the time Tε = T /ε 2 with T > 0 fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the NKGE with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with O(ε 2 ) wavelength in time and O(ε −2 ) wave speed, which indicates that the temporal error is independent of ε when the time step size is chosen as O(ε 2 ). Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.

Uniform Error Bounds of an Energy-Preserving Exponential Wave Integrator Fourier Pseudo-Spectral Method for the Nonlinear Schrödinger Equation with Wave Operator and Weak Nonlinearity

“…Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention. The long-time dynamics of the Klein-Gordon (KG) equations and Dirac equations with weak nonlinearity or small potential are thoroughly studied in the literature [7,8,18,[20][21][22][30][31][32]. For the weak nonlinear NLSW with periodic boundary condition, an exponential wave integrator Fourier pseudo-spectral method has been proposed in [24] and proved to be uniformly accurate about ε up to the time at O(1/ε 2 ).…”

Structure‐preserving exponential wave integrator methods and the long‐time convergence analysis for the Klein‐Gordon‐Dirac equation with the small coupling constant

“…Recently, the numerical methods for long-time dynamics of partial differential equations (PDEs) with weak nonlinearity (or small potentials) have received more and more attention. The long-time dynamics of the KG equations and Dirac equations with weak nonlinearity or small potential are thoroughly studied in the literature [11][12][13][14][22][23][24][25]. For long time analysis of other types of equations, we refer to Reference [46,47].…”

Improved error estimates of the time‐splitting methods for the long‐time dynamics of the Klein–Gordon–Dirac system with the small coupling constant

“…Furthermore, based on the RCO technique, improved uniform error bounds for Klein-Gordon equations (KGE) and Dirac/nonlinear Dirac equations (NLDE) have also been established [2,4,7]. The RCO technique has also been used for analyzing other types of PDEs such as sine-Gordon equation and space fractional KGE [20,25] as well as other numerical methods such as EWIFP methods and low regularity methods [19,25].…”