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A discrete action principle for electrodynamics and the construction of explicit symplectic integrators for linear, non-dispersive media. (English) Zbl 1162.78306

Summary: We derive a discrete action principle for electrodynamics that can be used to construct explicit symplectic integrators for Maxwell’s equations. Different integrators are constructed depending on the choice of discrete Lagrangian used to approximate the action. By combining discrete Lagrangians in an explicit symplectic partitioned Runge-Kutta method, an integrator capable of achieving any order of accuracy is obtained. Using the von Neumann stability analysis, we show that the integrators greatly increase the numerical stability and reduce the numerical dispersion compared to other methods. For practical purposes, we demonstrate how to implement the integrators using many features of the finite-difference time-domain method. However, our approach is also applicable to other spatial discretizations, such as those used in finite element methods. Using this implementation, numerical examples are presented that demonstrate the ability of the integrators to efficiently reduce and maintain a minimal amount of numerical dispersion, particularly when the time-step is less than the stability limit. The integrators are therefore advantageous for modeling large, inhomogeneous computational domains.

MSC:

78M20 Finite difference methods applied to problems in optics and electromagnetic theory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65Z05 Applications to the sciences
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
78M25 Numerical methods in optics (MSC2010)
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