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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*}
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with permittivity
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{latex} $\epsilon = \epsilon_0 \epsilon_r ${latex} |
and permeability
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$\mu = \mu_0 \mu_r$
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. In the current implementation, a constant value of the effective conductivity
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$\sigma_{eff} = \tan \delta \cdot 2 \pi f \cdot \epsilon $
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is assumed by fixing a frequency
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$f$ {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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\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*}
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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.