A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein–Gordon–Schrödinger equations in the nonrelativistic limit regime
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- 作者：Weizhu Bao ; Xiaofei Zhao
- 关键词：Mathematics Subject Classification65L05 ; 65L20 ; 65L70
- 刊名：Numerische Mathematik
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：135
- 期：3
- 页码：833-873
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
- 出版者：Springer Berlin Heidelberg
- ISSN：0945-3245
- 卷排序：135
文摘
A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein–Gordon–Schrödinger (KGS) equations in the nonrelativistic limit regime with a dimensionless parameter \(0<\varepsilon \le 1\) which is inversely proportional to the speed of light. In fact, the solution of the KGS equations propagates waves with wavelength at \(O(\varepsilon ^2)\) and O(1) in time and space, respectively, when \(0<\varepsilon \ll 1\), 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 \(O(\tau ^2/\varepsilon ^2+h^{m_0})\) and \(O(\varepsilon ^2+h^{m_0})\) for \(\varepsilon \in (0,1]\) with \(\tau \) time step size, h mesh size and \(m_0\ge 4\) 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 \(O(\tau )\) for \(\varepsilon \in (0,1]\). In addition, the MTI-FP method converges optimally with quadratic convergence rate at \(O(\tau ^2)\) in the regime when \(0<\tau \lesssim \varepsilon ^2\) and the error is at \(O(\varepsilon ^2)\) independent of \(\tau \) in the regime when \(0<\varepsilon \lesssim \tau ^{1/2}\). Thus the meshing strategy requirement (or \(\varepsilon \)-scalability) of the MTI-FP is \(\tau =O(1)\) and \(h=O(1)\) for \(0<\varepsilon \ll 1\), 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.