Part I. Numerical experiments for the computation of invariant curves in dynamical systems. Part II. Numerical convergence results for a one-dimensional Stefan problem

Citation

Morlet, Anne Chantal (1990) Part I. Numerical experiments for the computation of invariant curves in dynamical systems. Part II. Numerical convergence results for a one-dimensional Stefan problem. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/nfr6-ca66. https://resolver.caltech.edu/CaltechETD:etd-05092007-080826

Abstract

Part I

We derive a model equation for the linearized equation of an invariant curve for a Poincare map. We discretize the model equation with a second-order and third-order finite difference schemes, and with a cubic spline interpolation scheme. We also approximate the solution of the model equation with a truncated Fourier expansion. We derive error estimates for the second-order and third-order finite difference schemes and for the cubic spline interpolation scheme. We numerically implement the four schemes we consider and plot some error curves.

Part II

We show for a one-dimensional Stefan problem, that the numerical solution converges to the solution of the continuous equations in the limit of zero meshsize and timestep. We discretize the continuous equations with a second-order finite difference scheme in space and Crank-Nicholson scheme in time. We derive error equations and we use L2 estimates to bound the error in terms of the truncation errors of the finite difference scheme. We confirm the analysis with numerical computations. We numerically prove that we have fourth-order convergence in space if we discretize the partial differential equations with a fourth-order scheme in space.

Thesis Files