  We investigate the approximate dynamics of several differential equations
when the solutions are restricted to a sparse subset of a given basis. The
restriction is enforced at every time step by simply applying soft thresholding
to the coefficients of the basis approximation. By reducing or compressing the
information needed to represent the solution at every step, only the essential
dynamics are represented. In many cases, there are natural bases derived from
the differential equations which promote sparsity. We find that our method
successfully reduces the dynamics of convection equations, diffusion equations,
weak shocks, and vorticity equations with high frequency source terms.
