## I Introduction

The simulation of high-frequency electromagnetic problems is often carried out in frequency domain. This choice is motivated by the linearity of the underlying governing equations. However, the solution of problems in frequency domain may require the resolution of very large linear systems of equations and this becomes particularly inconvenient for broadband simulations such that approximations like model order reduction are typically used, e.g. [slone03, Floch_2015aa, Paquay_2016aa]. The coupling with nonlinear time-dependent systems and the computation of transients are other cases where time-domain simulations outperform frequency-domain simulations.

On the other hand, the numerical complexity resulting from time-domain simulations may also become prohibitively expensive. Parallelization in ‘space’, e.g., matrix-vector multiplications corresponding to the application of the curl operator, using multicore architectures is well established in academic and industrial software environments

[CST_2016aa]. However, the parallelization efficiency eventually saturates with increasing number of cores depending on the memory bandwidth of the involved hardware. Time-domain parallelization is a promising extension to domain decomposition in space.The development and application of parallel-in-time methods dates back more
than 50 years, see [nievergelt1964parallel]. These methods can be direct
[christlieb2010parallel, gander2013paraexp] or iterative
[lions2001parareal, minion2011hybrid]. They can also be well suited for small
scale parallelization [miranker1967parallel, womble1990time] or large parallelization
[gander2013paraexp, minion2011hybrid]. Recently, the *Parareal method*
gained interest [lions2001parareal, gander2007analysis]

. In its initial version, Parareal was developed for large scale semi-discretized parabolic partial differential equations (PDEs). It involves the splitting of the time interval and the resolution of the governing ordinary differential equation (ODE) in parallel on each sub-interval using a fine propagator which can be any classical time-stepper with a fine time grid. A coarse propagator distributes the initial conditions for each sub-interval during the Parareal iterations. It is typically obtained by a time stepper with a coarse grid on the entire time interval. Parareal iterates the resolution of both the coarse and the fine problems until convergence.

Most parallel-in-time methods fail for hyperbolic problems. In the case of Parareal,
analysis has shown that it may lead to the *beating* phenomenon depending on
the structure of the system matrix [farhat2006beat]

. It may even become unstable if the eigenvalues of the matrix are purely imaginary which is the case in the presence of undamped electromagnetic waves.

In this paper we apply the *ParaExp method* from [gander2013paraexp] for
the parallelization of time-domain resolutions of hyperbolic equations that govern
the electromagnetic wave problems as initially proposed in [merkel2016].

The method splits the time interval into sub-intervals and solves smaller problems on each sub-interval as visualized in Figure 1. Using the theory of linear ordinary differential equations, the total solution for each sub-interval is decomposed into particular solution with zero initial conditions and homogeneous solutions with initial conditions from previous intervals.

The paper is organized as follows: in Section II we introduce Maxwell’s equations and derive the governing system of ODEs for the wave equation obtained by the Finite Integration Technique (FIT). This system is then used in Section LABEL:sec:paraexp for the presentation of the ParaExp method following the lines of [gander2013paraexp]. The mathematical framework is briefly sketched and the details of the algorithm are discussed. The combination of ParaExp with Leapfrog is proposed. Section LABEL:sec:applications deals with numerical examples. We consider two applications: a simple wave guide problem and a realistic spiral inductor model discretized by the Finite Integration Technique. The examples are investigated in terms of efficiency, energy conservation and frequency spectrum.

## Ii Space and Time Discretization of Maxwell’s equations

In an open, bounded domain and , the evolution of electromagnetic fields is governed by Maxwell’s equations on , see e.g. [jackson1999classical]:

(1) | ||||||

(2) |

with suitable initial and boundary conditions at time and , respectively. In presence of linear materials, these equations are completed by constitutive laws [jackson1999classical]:

(3) |

In these equations, is the magnetic field [A/m], the magnetic flux density [T], the electric field [V/m], the electric flux density [C/m], , , and are the total, Ohmic, displacement and electric source current densities [A/m], is the electric charge density [C/m]. The material properties , and are the electric conductivity, the electric permittivity and the magnetic permeability, respectively. In this paper, we consider electromagnetic wave propagation in non-conducting media which are free of charges, i.e., and .

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