Full center manifold discretizations for near-onset convection patterns in the spherical Bénard problem

Large dynamical systems are often obtained as discretizations of parabolic PDEs with nonlinear elliptic parts, either equations or system of order 2 vor 2m, m>1. Space and time discretization methods, so called full discretizations, are necessary to determine the dynamics on center manifolds. We proved for the first time that, allowing stable, and center manifolds, for the standard space discretization methods, e.g. the standard methods used in nonlinear elliptic PDEs. the space discrete center manifolds converge to the original center manifolds: The coefficients of the Taylor expansion of a discrete center manifold and its normal form converge to those of the original center manifold. Then standard, e.g., Runge-Kutta, or geometric time discretization methods can be applied to the discrete center manifold system of small dimension of ordinary differential equations.

These results are applied to near-onset convection patterns in the spherical Benard problem in the earth mantle.This problem is 5-determined, so we need the center manifold, instead of a Liapunov-Schmidt technique. The numerical method has to inherit the equivariance, so that of the spherical harmonics. We use a Chebyshev collocation spectral method, and instead of the exact we obtain an approximate discrete normal form.