Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
6. Numerical experiment: solution of a convection problem
We consider the rotation of a scalar with a cone as initial condition and a sharp edged bottomline:

In the following, a comparison is given of the finite difference discretization on adaptive sparse grids and a similar  non-adaptive finite difference scheme on uniform grids. 
adaptive non-adaptive
convective terms 5th order WENO 5th order WENO
finest mesh size 1/512 1/64,.....1/512
refinement threshold and insert neighbours 
with various trehold values
time discretization 3rd oder  Adams-Bashforth 3rd order Adams-Bashforth
time step 1/5120 1/5120
wavelets 4th order Interpolets none

In this example refinement is required only near the bottom line of the cone which is a one-dimensional manifold. Therefore, the adaptive scheme achieves twice the convergence rate (error after one rotation vs. number of DOF) of the non-adaptive scheme.