A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein-Gordon-Schrodinger equations in the nonrelativistic limit regime

作者：Bao Weizhu; Zhao Xiaofei

来源：Numerische Mathematik, 2017, 135(3): 833-873.

DOI：10.1007/s00211-016-0818-x

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein-Gordon-Schrodinger (KGS) equations in the nonrelativistic limit regime with a dimensionless parameter which is inversely proportional to the speed of light. In fact, the solution of the KGS equations propagates waves with wavelength at and O(1) in time and space, respectively, when , which brings significantly numerical burden in practical computation. The MTI-FP method is designed by adapting a multiscale decomposition by frequency of the solution at each time step and applying the Fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives, respectively. We rigorously establish two independent error bounds for the MTI-FP at and for with time step size, h mesh size and an integer depending on the regularity of the solution, which imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at for . In addition, the MTI-FP method converges optimally with quadratic convergence rate at in the regime when and the error is at independent of in the regime when . Thus the meshing strategy requirement (or -scalability) of the MTI-FP is and for , which is significantly better than that of classical methods. Numerical results demonstrate that our error bounds are optimal and sharp. Finally, the MTI-FP method is applied to study numerically convergence rates of the KGS equations to its limiting models in the nonrelativistic limit regime.

• 出版日期2017-3