Numerical analysis and long-term dynamics of Fourier-type pseudo-spectral schemes applied to the Klein-Gordon equation.

Abstract

This thesis investigates the Klein-Gordon equation (KGE) using a combined theoretical and numerical approach. We develop a robust numerical approximation scheme for the KGE that demonstrates good convergence properties for various data types and explore the long-time behavior of semi-discrete KGE solutions near finite- and infinite-dimensional invariant subsets of an appropriate space. In the first part of the thesis, we establish convergence results for the semidiscrete, Fourier pseudo-spectral spatial approximation of the KGE with smooth potentials. We present an extensive stability and convergence analysis for finite Sobolev regularity data in T and R, as well as for smooth data from Gevrey classes in T. We demonstrate that the convergence rate is algebraic in the first case and (sub-)geometric in the second case. The second part of the thesis deals with the numerical studies of the long-time dynamics of semi-discrete numerical solutions in periodic settings. Through an extensive set of simulations, we show that the pseudospectral semi-discretization is capable of preserving finite- and infinite-dimensional invariant structures over very long time intervals.