We present a high-order, adaptive numerical method for numerically solving advection-diffusion PDEs, and demonstrate the advantage of this method for fluid dynamics problems, especially that generate quasi-singular or coherent structures.

There are four aspects of the method that we will elucidate. First, the spatial discretization employs piecewise-polynomial spaces similar to

the traditional spectral-element method; however we use orthogonal multiresolution analysis (MRA) comprising many nonconforming elements of various scales. This MRA is constructed using multiwavelets, which are orthonormal basis functions whose span efficiently describes the extra ``detail'' information gained in refining some of the spectral elements (Alpert et al. 1999).

Second, the adaptivity criterion for a set of elements is the norm of the projection onto the corresponding multiwavelets. Using this simple

criterion, the algorithm refines and coarsens the representation of steepening or translating structures. Our criterion replaces more traditional criteria, based on local gradients or local spectra.

Third, we take advantage of the spatial discretization, to improve the time-stepping scheme. We use the ``exact linear part'' scheme (Beylkin et al. 1998), which amounts to using scaling and squaring to compute the exponential propagator due to the diffusive linear operator. This adaptive operator exponentiation is made feasible by our spatial discretization. It significantly improves convergence for high Reynolds numbers and large time steps.

Finally, we use block-sparse data structures to achieve efficiency.

We will demonstrate these features using the standard shallow-water test suite of Williamson et al. (1992).