  We propose an integrable discrete model of one-dimensional soil water
infiltration. This model is based on the continuum model by Broadbridge and
White, which takes the form of nonlinear convection-diffusion equation with a
nonlinear flux boundary condition at the surface. It is transformed to the
Burgers equation with a time-dependent flux term by the hodograph
transformation. We construct a discrete model preserving the underlying
integrability, which is formulated as the self-adaptive moving mesh scheme. The
discretization is based on linearizability of the Burgers equation to the
linear diffusion equation, but the na\"ive discretization based on the Euler
scheme which is often used in the theory of discrete integrable systems does
not necessarily give a good numerical scheme. Taking desirable properties of a
numerical scheme into account, we propose an alternative discrete model that
produces solutions with similar accuracy to direct computation on the original
nonlinear equation, but with clear benefits regarding computational cost.
