  A weakly conducting liquid droplet immersed in another leaky dielectric
liquid can exhibit rich dynamical behaviors under the effect of an applied
electric field. Depending on material properties and field strength, the
nonlinear coupling of interfacial charge transport and fluid flow can trigger
electrohydrodynamic instabilities that lead to shape deformations and complex
dynamics. We present a spectral boundary integral method to simulate droplet
electrohydrodynamics in a uniform electric field. All physical variables, such
as drop shape and interfacial charge density, are represented using spherical
harmonic expansions. In addition to its exponential accuracy, the spectral
representation affords a nondissipative dealiasing method required for
numerical stability. A comprehensive charge transport model, valid under a wide
range of electric field strengths, accounts for charge relaxation, Ohmic
conduction, and surface charge convection by the flow. A shape
reparametrization technique enables the exploration of significant droplet
deformation regimes. For low-viscosity drops, the convection by the flow drives
steep interfacial charge gradients near the drop equator. This introduces
numerical ringing artifacts we treat via a weighted spherical harmonic
expansion, resulting in solution convergence. The method and simulations are
validated against experimental data and analytical predictions in the
axisymmetric Taylor and Quincke electrorotation regimes.
