Nonlinear Galerkin Methods for 3D Magnetohydrodynamic Equations

The usage of nonlinear Galerkin methods for the numerical solution of partial differential equations is demonstrated by treating an example. We describe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan–Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions.