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

Abstract

AbstractRecently, 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 , 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 up to the time at for where and are the mesh size and time step, respectively, and 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 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 any energy‐preserving exponential wave integrator method and any relevant long time analysis for the KGD equation.