  We develop a numerical algorithm for computing the effective drift and
diffusivity of the steady-state behavior of an overdamped particle driven by a
periodic potential whose amplitude is modulated in time by multiplicative noise
and forced by additive Gaussian noise (the mathematical structure of a flashing
Brownian motor). The numerical algorithm is based on a spectral decomposition
of the solution to the Fokker-Planck equation with periodic boundary conditions
and the cell problem which result from homogenization theory. We also show that
the numerical method of Wang, Peskin, Elston (WPE, 2003) for computing said
quantities is equivalent to that resulting from homogenization theory. We show
how to adapt the WPE numerical method to this problem by means of discretizing
the multiplicative noise via a finite-volume method into a discrete-state
Markov jump process which preserves many important properties of the original
continuous-state process, such as its invariant distribution and detailed
balance. Our numerical experiments show the effectiveness of both methods, and
that the spectral method can have some efficiency advantages when treating
multiplicative random noise, particularly with strong volatility.
