In 1831, Faraday observed the formation of standing waves on the surface of a vibrating fluid body. Subsequent experiments have revealed the existence of a rich tapestry of patterned states that can be accessed by varying the frequency and amplitude of the vibration and have spurred vast research in hydrodynamics and pattern formation. These include linear analyses to determine the conditions for the onset of the patterns, weakly nonlinear studies to understand pattern selection and dynamical systems approaches to study mode competition and chaos. Recently, there has been some work towards numerical simulations in various three-dimensional geometries. These methods however possess low orders of accuracy, making them unsuitable for nonlinear regimes.
We present a technique for fast and accurate simulations of nonlinear Faraday waves in a cylinder. Beginning from the viscous potential flow model of Dias et al, we generalize the Transformed Field Expansion to this geometry for finding the highly non-local Dirichlet-to-Neumann map for the Laplace equation. A spectral method relying on Zernike polynomials is developed to rapidly and accurately compute the bulk potential. The free surface evolution equations are solved in time using fourth-order Runge-Kutta and refined to a high order by Picard iterations. The results are in perfect agreement with the instability thresholds and surface patterns predicted for the linearized problem. The nonlinear simulations reproduce several qualitative features observed experimentally. In addition, by enabling one to switch between various nonlinear regimes, the technique allows a precise determination of the mechanisms triggering various experimental observations.