Third order WENO and CWENO reconstruction are widespread high order
reconstruction techniques for numerical schemes for hyperbolic conservation and
balance laws. In their definition, there appears a small positive parameter,
usually called ϵ, that was originally introduced in order to avoid a
division by zero on constant states, but whose value was later shown to affect
the convergence properties of the schemes. Recently, two detailed studies of
the role of this parameter, in the case of uniform meshes, were published. In
this paper we extend their results to the case of finite volume schemes on
non-uniform meshes, which is very important for h-adaptive schemes, showing the
benefits of choosing ϵ as a function of the local mesh size hj. In
particular we show that choosing ϵ=hj2 or ϵ=hj is
beneficial for the error and convergence order, studying on several non-uniform
grids the effect of this choice on the reconstruction error, on fully discrete
schemes for the linear transport equation, on the stability of the numerical
schemes. Finally we compare the different choices for ϵ in the case of
a well-balanced scheme for the Saint-Venant system for shallow water flows and
in the case of an h-adaptive scheme for nonlinear systems of conservation laws
and show numerical tests for a two-dimensional generalisation of the CWENO
reconstruction on locally adapted meshes