  A weighted residual collocation methodology for simulating two-dimensional
shear-driven and natural convection flows has been presented. Using a dyadic
mesh refinement, the methodology generates a basis and a multiresolution scheme
to approximate a fluid flow. To extend the benefits of the dyadic mesh
refinement approach to the field of computational fluid dynamics, this article
has studied an iterative interpolation scheme for the construction and
differentiation of a basis function in a two-dimensional mesh that is a finite
collection of rectangular elements. We have verified that, on a given mesh, the
discretization error is controlled by the order of the basis function. The
potential of this novel technique has been demonstrated with some
representative examples of the Poisson equation. We have also verified the
technique with a dynamical core of two-dimensional flow in primitive variables.
An excellent result has been observed-on resolving a shear layer and on the
conservation of the potential and the kinetic energies with respect to
previously reported benchmark simulations. In particular, the shear-driven
simulation at CFL = 2.5 (Courant-Friedrichs-Lewy) and $\mathcal Re = 1\,000$
(Reynolds number) exhibits a linear speedup of CPU time with an increase of the
time step, $Delta t$. For the natural convection flow, the conversion of the
potential energy to the kinetic energy and the conservation of total energy is
resolved by the proposed method. The computed streamlines and the velocity
fields have been demonstrated.
