  In this paper we extend the recently developed third-order limiter function
$H_{3\text{L}}^{(c)}$ [J. Sci. Comput., (2016), 68(2), pp.~624--652] to make it
applicable for more elaborate test cases in the context of finite volume
schemes. This work covers the generalization to non-uniform grids in one and
two space dimensions, as well as two-dimensional Cartesian grids with adaptive
mesh refinement (AMR). The extension to 2D is obtained by the common approach
of dimensional splitting. In order to apply this technique without loss of
third-order accuracy, the order-fix developed by Buchm\"uller and Helzel [J.
Sci. Comput., (2014), 61(2), pp.~343--368] is incorporated into the scheme.
Several numerical examples on different grid configurations show that the
limiter function $H_{3\text{L}}^{(c)}$ maintains the optimal third-order
accuracy on smooth profiles and avoids oscillations in case of discontinuous
solutions.
