In the field of Scientific Computing there is a big focus on solving time dependent Partial Differential Equations (PDEs) as efficiently and fast as possible. In order to do so, the PDE is discretized and solved on a mesh at every time step. Adaptive mesh refinement is used to develop a mesh at every time step which is sparse and which results in an accurate approximation to the solution. Interpolating wavelets are successfully used in adaptive mesh refinement (AMR). A detailed comparison of two wavelets for AMR is done on different data sets: Donoho’s interpolating wavelet and a lifted version (also called second generation wavelets) of Donoho’s interpolating wavelet. Moreover, various ways of handling the boundaries are considered. An algorithm to construct the meshes using wavelets is tested and optimized. Donoho’s interpolating wavelet with lower order boundary stencil implementation appears to be the most accurate, whilst resulting in very high compression compared to the original mesh. Furthermore, adapting the algorithm which constructs the meshes such that it adds more points for very irregular shapes, turns out to be valuable for solutions with fast changing features. For one such PDE, Donoho’s interpolating wavelet keeps less than 5% of the points whilst having an error smaller than 10􀀀4, in other words a sparsification of 20 times. Lastly, an improvement on the inverse transform during the adaptive mesh refinement leads to promising results. @en