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Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime. (English) Zbl 1453.65278

Summary: Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter \(\varepsilon \in(0, 1],\) which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. \(0 < \varepsilon \ll 1\), the solution of the NKGE propagates waves with wavelength at \(O(1)\) and \(O(\varepsilon^2)\) in space and time, respectively, which brings significantly numerical burdens in designing numerical methods. We compare systematically spatial/temporal efficiency and accuracy as well as \(\epsilon\)-resolution (or \(\epsilon\)-scalability) of different numerical methods including finite difference time domain methods, time-splitting method, exponential wave integrator, limit integrator, multiscale time integrator, two-scale formulation method and iterative exponential integrator. Finally, we adopt the multiscale time integrator to study the convergence rates from the NKGE to its limiting models when \(\varepsilon \to 0^+\).

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
65Z05 Applications to the sciences
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