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## Key

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In our approach, Ampere's and Faraday's laws are combined and integrated over time to yield the inhomogeneous vector wave equation for the time integral of the electric field E:

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{latex}\begin{eqnarray*}\left( \epsilon \frac{\partial^2}{\partial t^2} + \sigma_{eff} \frac{\partial}{\partial t} + \nabla \times \mu^{-1}\nabla\times \right) \int^t \emph{E} d \tau = -\emph{J}
\end{eqnarray*}
{latex}

with permittivity

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{latex} $\epsilon = \epsilon_0 \epsilon_r${latex}
and permeability
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{latex}
$\mu = \mu_0 \mu_r$
{latex}
. In the current implementation, a constant value of the effective conductivity
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{latex}
$\sigma_{eff} = \tan \delta \cdot 2 \pi f \cdot \epsilon$

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is assumed by fixing a frequency
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{latex}
$f$ {latex}
, and the losses are specified by the loss tangent
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{latex} $\tan \delta$ {latex}
. As is common for Wakefield computations of rigid beams, the electric current source density J is given by a one-dimensional Gaussian particle distribution, moving at the speed of light along the beam line.

The computational domain is discretized into curved tetrahedral elements and

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{latex} $\int^t \emph{E}d \tau${latex}
in Eq. (1) is represented as an expansion in hierarchical Whitney vector basis functions
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{latex}$\emph{N}_i (x)${latex}

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{latex}
\begin{eqnarray*} \int^t \emph{E} (\emph{x}, \tau) d \tau = \sum^{N_p}_{i=1} e_i (t) \cdot \emph{N}_i (x) \end{eqnarray*}

{latex}

up to order

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{latex} $p$ {latex}
within each element. For illustration, the numbers of basis functions for first, second and sixth order are
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{latex} $N_1 = 6${latex}
,
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{latex} $N_2 = 20${latex}
and
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{latex} $N_6 = 216${latex}
, respectively.  Higher order elements (both curved and with higher-order basis functions) not only significantly improve field accuracy and dispersive properties, but also generally lead to higher-order accurate particle-field coupling equivalent to, but much less laborious than, complicated higher-order interpolation schemes commonly found in finite-difference methods.