Numerical Solution of a Nonlinear Dissipative System Using a Pseudospectral Method and Inertial Manifolds

We consider the numerical solution of nonlinear, dissipative partial differential equations using a pseudospectral method and the methodology of approximate inertial manifolds. Coarse and fine grids are employed, with a nonlinear mapping to relate the solutions computed on these grids. The approach is illustrated by consideration of the Kuramoto-Sivashinsky equation subject to periodic solutions that are restricted to the invariant subspace of odd functions. Numerical results are presented on the ability of the method to mimic the time evolution of the infinite-dimensional dynamical system. The results demonstrate that the coarse/fine grid method is more accurate than the standard pseudospectral method on the same coarse grid. However, time integration studies suggest that the increased accuracy is bought at an increase in computational cost: to achieve a specified accuracy, the computational cost of the two-grid method exceeds that of a standard pseudospectral method for this spatially periodic problem.