To cite this version:Stéphane Balac, Fabrice Mahé. Embedded Runge-Kutta scheme for step-size control in the interaction picture method. Computer Physics Communications, Elsevier, 2013, 184 (4) AbstractWhen solving certain evolution type PDE such as the Schrödinger equation, the Interaction Picture method is a valuable alternative to Split-Step methods. The Interaction Picture method has good computational features when used together with the standard 4th order Runge-Kutta scheme (giving rise to the RK4-IP method). In this paper we present an embedded Runge-Kutta scheme with orders 3 and 4with the aim to deliver an estimation of the local error for adaptive step-size control purposes in the Interaction Picture method. The corresponding ERK4(3)-IP method preserves the features of the RK4-IP method and provide a local error estimate at no significant extra cost.
Abstract. The "interaction picture" (IP) method is a very promising alternative to Split-Step methods for solving certain type of partial differential equations such as the nonlinear Schrödinger equation involved in the simulation of wave propagation in optical fibers. The method exhibits interesting convergence properties and is likely to provide more accurate numerical results than cost comparable Split-Step methods such as the Symmetric Split-Step method. In this work we investigate in detail the numerical properties of the IP method and carry out a precise comparison between the IP method and the Symmetric Split-Step method.
In optics the nonlinear Schrödinger equation (NLSE) which modelizes wave propagation in an optical fiber is mostly solved by the Symmetric Split-Step method. The practical efficiency of the Symmetric Split-Step method is highly dependent on the computational grid points distribution along the fiber, therefore an efficient adaptive step-size control strategy is mandatory. The most common approach for step-size control is the "step-doubling" approach. It provides an estimation of the local error for an extra computational cost of around 50 %. Alternatively there exist in optics literature other approaches based on the observation along the propagation length of the behavior of a given optical quantity. The step-size at each computational step is set so as to guarantee that the known properties of the quantity are preserved. These approaches derived under specific physical assumptions are low cost but suffer from a lack of generality. In this paper we present a new method for estimating the local error in the Symmetric Split-Step method when solving the NLSE. It conciliates the advantages of the step-doubling approach in terms of generality without the drawback of requiring a significant extra computational cost. The method is related to Embedded Split-Step methods for nonlinear evolution problems.
Whispering gallery modes [WGM] are resonant modes displaying special features: they concentrate along the boundary of the optical cavity at high polar frequencies and they are associated with complex scattering resonances very close to the real axis. As a classical simplification of the full Maxwell system, we consider 2D Helmholtz equations governing transverse electric or magnetic modes. Even in this 2D framework, very few results provide asymptotic expansion of WGM resonances at high polar frequency $m\to \infty $ for cavities with radially varying optical index. In this work, using a direct Schrödinger analogy, we highlight three typical behaviors in such optical micro-disks, depending on the sign of an ‘effective curvature’ that takes into account the radius of the disk and the values of the optical index and its derivative. Accordingly, this corresponds to abruptly varying effective potentials (step linear or step harmonic) or more classical harmonic potentials, leading to three distinct asymptotic expansions for ground state energies. Using multiscale expansions, we design a unified procedure to construct families of quasi-resonances and associate quasi-modes that have the WGM structure and satisfy eigenequations modulo a super-algebraically small residual ${\mathscr{O}}(m^{-\infty })$. We show using the black box scattering approach that quasi-resonances are ${\mathscr{O}}(m^{-\infty })$ close to true resonances.
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