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

Abstract

Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity (or small potentials) have received more and more attention. For the Klein-Gordon-Dirac (KGD) equation with the small coupling constant 𝜀 ∈ (0, 1], we propose two time symmetric and structure-preserving exponential wave integrator Fourier pseudo-spectral (TSSPEWIFP) methods under periodic boundary conditions. The new methods are proved to preserve the energy in discrete level and in addition the first one preserves the modified discrete mass. Through rigorous error analysis, we establish uniform error bounds of the numerical solutions at O(h m 0 + 𝜀 1−𝛽 𝜏 2 ) up to the time at O(1∕𝜀 𝛽 ) for 𝛽 ∈ [0, 1] where h and 𝜏 are the mesh size and time step, respectively, and m 0 depends on the regularity conditions. Compared with the results of existing numerical analysis, our analysis has the advantage of showing the long time numerical errors for the KGD equation with the small coupling constant. The tools for error analysis mainly include cut-off technique and the standard energy method. We also extend the results on error bounds, structure-preservation and time symmetry to the oscillatory KGD equation with wavelength at O(𝜀 𝛽 ) in time which is equivalent to the KGD equation with small coupling constant. Numerical experiments confirm that the theoretical results are correct. Our methods are novel because that to the best of our knowledge there has not been

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 ).…”

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

“…This suggests that the FD and CFD methods are under-resolution in both space and time with respect to 𝜀 ∈ (0, 1]. Recently, for the KGDS (1.1), two structure-preserving EWIFP methods have been proposed [30]. However, the errors of the methods can only be proved to have uniform bounds O(h m + 𝜏 2 ) rather than improved bounds O(h m + 𝜀𝜏 2 ) up to the time at O(1∕𝜀).…”

“…This suggests that the FD and CFD methods are under‐resolution in both space and time with respect to $$$ \varepsilon \in \left(0,1\right] $$$. Recently, for the KGDS (1.1), two structure‐preserving EWIFP methods have been proposed [30]. However, the errors of the methods can only be proved to have uniform bounds $$$ O\left({h}^m+{\tau}^2\right) $$$ rather than improved bounds $$$ O\left({h}^m+\varepsilon {\tau}^2\right) $$$ up to the time at $$$ O\left(1/\varepsilon \right) $$$.…”