An approximate inertial manifold for computing Burgers' equation

We present a numerical scheme for the approximation of nonlinear evolution equations over large time intervals. Our algorithm is motivated from the dynamical systems point of view. In particular, we adapt the methodology of approximate inertial manifolds to a finite difference scheme. This leads to a differential treatment in which the higher (i.e. unresolved) modes are expressed in terms of the lower modes. As a particular example we derive an approximate inertial manifold for Burgers' equation and develop a numerical algorithm suitable for computing. We perform a parameter study in which we compare the accuracy of a standard scheme with our modified scheme. For all values of the parameters (which are the coefficient of viscosity and the cell size), we obtain a decrease in the numerical error by at least a factor of 2.0 with the modified scheme. The decrease in error is substantially greater over large regions of the parameters space.