Coarse projective integration for low-dimensional turbulence models

A new method from the Equation-free Multiscale framework, Coarse Projective Integration (CPI), is applied to three different low-dimensional turbulence models. The first model is the (one-dimensional) randomly-forced Burgers equation, the second is the (also one-dimensional) MMT equation, a non-linear dispersive model for wave turbulence, and the last is the hyperviscous formulation of the Navier-Stokes equation in two-dimensions. The goal of the application of CPI to the computations is to reduce the computational (CPU or wall clock time) time T required to evolve the system from its initial state, S, to its state at time tau, S tau. The application of CPI to the Burgers equation did not result in a significant saving in T. However, application of CPI to the MMT equation and the 2D Navier-Stokes equation did result in significant savings in computational time for both equations. The savings, under particular conditions discussed in the text, were a factor of 3.74 for the MMT equation, and a factor of 11.87 for the 2D Navier-Stokes equation. The requirements for the system S for CPI to be useful are discussed. The saving in computational time that would result from applying CPI to Navier-Stokes turbulence in three dimensions is estimated.