  We develop a finite-dimensional approximation of the Frobenius-Perron
operator using the finite volume method applied to the continuity equation for
the evolution of probability. A Courant-Friedrichs-Lewy condition ensures that
the approximation satisfies the Markov property, while existing convergence
theory for the finite volume method guarantees convergence of the discrete
operator to the continuous operator as mesh size tends to zero. Properties of
the approximation are demonstrated in a computed example of sequential
inference for the state of a low-dimensional mechanical system when
observations give rise to multi-modal distributions.
