  Bayesian estimation strategies represent the most fundamental formulation of
the state estimation problem available, and apply readily to nonlinear systems
with non-Gaussian uncertainties. The present paper introduces a novel method
for implementing grid-based Bayesian estimation which largely sidesteps the
severe computational expense that has prevented the widespread use of such
methods. The method represents the evolution of the probability density
function (PDF) in phase space, $p_{\x}(\x',t)$, discretized on a fixed
Cartesian grid over {\it all} of phase space, and consists of two main steps:
(i) Between measurement times, $p_{\x}(\x',t)$ is evolved via numerical
discretization of the Kolmogorov forward equation, using a Godunov method with
second-order corner transport upwind correction and a total variation
diminishing flux limiter; (ii) at measurement times, $p_{\x}(\x',t)$ is updated
via Bayes' theorem. Computational economy is achieved by exploiting the
localised nature of $p_{\x}(\x',t)$. An ordered list of cells with
non-negligible probability, as well as their immediate neighbours, is created
and updated, and the PDF evolution is tracked {\it only} on these active cells.
%The grid-based discretization of $p_{\x}(\x',t)$ in this approach avoids the
requirement for resampling associated with particle-based representations of
the PDF.
