Adaptive Wavelet Methods


Many partial differential equations have solutions that are localized in both position and time. For example, turbulence is characterized by highly intermittent and non-stationary vorticity fields. In addition, high Reynolds number turbulence has motions over a very large range of length scales and thus requires a very fine computational grid and small time step. Other examples incluse combustion, chemical reactions, fluid-structure interaction, and shock propagation. Because of the fine resolution required, conventional techniques are not efficient for these problems. We require a method where the computational grid automatically adapts match the solution. In other words, we want to put the grid points only where they are needed.

Our adaptive wavelet method is based on tensor products of second generation wavelets constructed on a hierarchy of nested dyadic grids. Because we use collocation wavelets, each wavelet (at a given position and scale) corresponds to a particular grid point. Therefore, by transforming the solution to wavelet space and then removing wavelets with coefficients smaller than a given threshold we automatically adapt the grid to the solution.

We have been using adaptive wavelet filtering for the following problems:

1. Numerical solution of time evolution problems. To allow for the change in the solution over a time step, we add nearest neighbour grid points in position and scale. The nearest neighbours in position allow for advection, and the nearest neighbours in scale avoid aliasing errors. We have applied this method to solve turbulence modelling and fluid-structure interaction problems.

2. Simultaneous space-time method. In this case we solve a time evolution problem on a space-time domain. This allows us to adapt both the spatial resolution and time step to local intermittency. We are applying this method to the Burgers equation and the Kuramoto-Sivashinsky equation in one space dimension, and intend to apply it to 2D and 3D turbulence problems.

3. Signal analysis. We have used the de-noising properties of wavelet filtering to extract the coherent structures (i.e. vortices) from turbulent flows. This method can also be used as the filter in a LES-like turbulence model.

4. Solution of elliptic partial differential equations. We have developed an adaptive wavelet version of the multilevel method for linear and nonlinear partial differential equations.





Jahrul Alam



  Marie Farge (Ecole Normale Superieure, Paris, France)
  Daniel Goldstein (University of Colorado at Boulder, USA)
  Kai Schneider (Universite de Marseille, France)

Oleg V. Vasilyev

(University of Colorado at Boulder, USA)

Engineering Collaborator(s):


Samir Ziada

(Department of Mechanical Engineering, McMaster University)


  • An adaptive wavelet collocation method for fluid--structure interaction at high Reynolds numbers
  • Hybrid wavelet collocation-Brinkman penalization method for complex geometry flows
  • Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis
  • Vorticity filaments in two-dimensional turbulence: creation, stability and effect