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## Efficient high order time schemes for Maxwell's equations

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Hélène Barucq
Marc Duruflé

#### Abstract

The solution of time domain Maxwell's equations with locally time-stepping is of particular interest in case of realistic applications for which local high-order space approximations or locally refined mesh are relevant. The general form of the corresponding ODE (Ordinary Differential Equations) reads: $M_h dU/dt + K_h U = F (t)$, where M_h is the mass matrix and $K_h$ the stiffness matrix. This is a stiff problem because the eigenvalues of $M_h^{ −1} K_h$ may be large. As a consequence, the stability condition of explicit time schemes becomes so restrictive that it turns out interesting to use locally implicit time discretization to release the stability constraint in given regions. As a preliminary work for the design of high order locally implicit discretizations of wave problems, we provide a performance assessment of different implicit schemes. We focus on one-step methods mainly the classical implicit Runge-Kutta (SDIRK, DIRK, e.g. see [1]) schemes and Padé schemes (see [2]). The comparison criteria are based on the amplitude and phase errors which are reliable gauges of accuracy when approximating waves problems. Regarding Padé schemes, handling the source term F deserves a particular attention to keep the level of accuracy of high order discretizations. Numerical experiments are conducted in 1-D and 2-D for Maxwell's equations. The numerical results are performed with high order finite elements by using the code Montjoie (see [3]).

### Dates and versions

hal-01403636 , version 1 (01-12-2016)

### Identifiers

• HAL Id : hal-01403636 , version 1

### Cite

Hélène Barucq, Marc Duruflé, Mamadou N'Diaye. Efficient high order time schemes for Maxwell's equations. ICOSAHOM 2016 - International Conference On Spectral and High Order Methods, Jun 2016, Rio de Janeiro, Brazil. ⟨hal-01403636⟩

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