  The multiscale complexity of modern problems in computational science and
engineering can prohibit the use of traditional numerical methods in
multi-dimensional simulations. Therefore, novel algorithms are required in
these situations to solve partial differential equations (PDEs) with features
evolving on a wide range of spatial and temporal scales. To meet these
challenges, we present a multiresolution wavelet algorithm to solve PDEs with
significant data compression and explicit error control. We discretize in space
by projecting fields and spatial derivative operators onto wavelet basis
functions. We provide error estimates for the wavelet representation of fields
and their derivatives. Then, our estimates are used to construct a sparse
multiresolution discretization which guarantees the prescribed accuracy.
Additionally, we embed a predictor-corrector procedure within the temporal
integration to dynamically adapt the computational grid and maintain the
accuracy of the solution of the PDE as it evolves. We present examples to
highlight the accuracy and adaptivity of our approach.
