High‐order, leapfrog methodology for the temporally dependent Maxwell's equations

Abstract

Abstract.A high-order, leapfrog integrator is presented and analyzed for Maxwell's time domain equations. The integrator is shown to be quite robust in terms of its efficiency and memory demands. Moreover, the integrator retains the original simplicity of the traditional second-order, leapfrog integrator. Rigorous Fourier analyses are presented to quantify the dispersion, dissipation, and stability properties of the integrator. To limit the scope of the presentation, the 2x2, 2x4, 4x2, and 4x4 schemes are examined and compared. Here the first digit denotes temporal accuracy; the second digit denotes spatial accuracy. Numerical demonstrations, using the three-dimensional rectangular waveguide as the object under test, are provided to further substantiate the theoretical analysis.

High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

“…In an attempt to offer an alternative to Runge-Kutta schemes, we shall use family of high-order explicit leap-frog (LF) schemes originally proposed by Young [21]. The chief attributes of these integrators are that the memory requirements are small and the algorithmic complexity is not significantly increased, with respect to the second-order leap-fi"og scheme.…”

Analysis of a high-order space and time discontinuous Galerkin method for elastodynamic equations. Application to 3D wave propagation

“…Note that linear combinations (12) are still valid with superscripts. According to Young [62], the leap-frog integration scheme with an order N (even) is written:…”

“…Indeed, unlike the ADER method that has the same level of accuracy in space and time, the standard leap-frog time scheme, second-order accurate, reduces the global convergence order limiting the interest for high-order space discretizations. Then, we propose an extension of the leap-frog scheme to higher (even) orders of accuracy, following a method proposed for the Maxwell equations by Young [62] and applied in the DG context by Fahs [29]. This method allows us to achieve temporal accuracy to any even order desired, when free surface conditions are applied, by introducing an iterative procedure.…”

Elastic Wave Propagation

“…But, since it is second order accurate, the global accuracy of the scheme INRIA can be penalized by the time approximation when higher-degree polynomials (m > 2) are used for spatial approximation. Then, we propose a higher-order leap-frog scheme following the method, proposed for the Maxwell equations, by Young [16] or Spachmann et al [14] and applied to a discontinuous Galerkin method by Fahs [4]. For a detailed description of the method, we introduce a simplified two equations problem whose unknowns are v(x, t) and σ(x, t)…”

“…According to the first results of the method presented in [3], the time accuracy of the scheme is crucial when global high-order accuracy is required. Then, we propose an extension of the leap-frog scheme to higher orders of accuracy following a method proposed for the Maxwell equations by Young [16] or Spachmann et al [14] and applied to DG methods by Fahs [4]. This method allows us to achieve temporal accuracy to any even order desired by introducing an iterative procedure.…”