In the field of Scientific Computing there is a big focus on solving time dependent Partial Differential Equations (PDEs) as efficiently as possible. Adaptive mesh refinement (AMR) can be used to construct a sparse mesh at every time step which maintains an accurate approximation to the solution. Interpolating wavelets are often used in AMR. In this report we present a detailed comparison of two wavelets for AMR: Donoho's interpolating wavelet and a lifted version (also called second generation wavelets) of Donoho's interpolating wavelet. The wavelets are compared on PDE problems from computational finance and computational fluid dynamics. We also examine different ways of handling the boundaries and the impact thereof. 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. For one data set Donoho's interpolating wavelet keeps fewer than 5% of the points whilst having an error smaller than 0.0001. In general, Donoho's interpolating wavelet produces sparse meshes while maintaining good accuracy, even for very irregular shapes. Lastly, an improvement on the inverse transform during the adaptive mesh refinement leads to promising results.