  A nonlinear adaptive procedure for optimising both the schemes in time and
space is proposed in view of increasing the numerical efficiency and reducing
the computational time. The method is based on a four-parameter family of
schemes we shall tune in function of the physical data (velocity, diffusion),
the characteristic size in time and space, and the local regularity of the
function leading to a nonlinear procedure. The \textit{a posteriori} strategy
we adopt consists in, given the solution at time $t^n$, computing a candidate
solution with the highest accurate schemes in time and space for all the nodes.
Then, for the nodes that present some instabilities, both the schemes in time
and space are modified and adapted in order to preserve the stability with a
large time step. The updated solution is computed with node-dependent schemes
both in time and space. For the sake of simplicity, only convection-diffusion
problems are addressed as a prototype with a two-parameters five-points finite
difference method for the spatial discretisation together with an explicit time
two-parameters four-stages Runge-Kutta method. We prove that we manage to
obtain an optimal time-step algorithm that produces accurate numerical
approximations exempt of non-physical oscillations.
