ERROR BOUNDS OF COMPACT FINITE DIFFERENCE METHODS FOR SOME DISPERSIVE PDEs AND APPLICATIONS

ZHANG TENG

Publication

ERROR BOUNDS OF COMPACT FINITE DIFFERENCE METHODS FOR SOME DISPERSIVE PDEs AND APPLICATIONS

ZHANG TENG

Abstract

The aim of this thesis is to propose and analyze some fourth-order compact finite difference schemes (4cFDs) for approximating several highly oscillatory dispersive PDEs, including the nonlinear Klein-Gordon equation in the nonrelativistic regime and the Zakharov system in the subsonic regime. Proofs of error estimate based on energy methods and cut-off techniques are presented, and numerical results are reported for verification purposes. Conservative schemes and schemes with uniform error bounds are considered. Finally, we apply the 4cFD to discretize Laplace’s equation with nonstandard boundary conditions for preparing the periodic initial data in simulations of quantized vortex interactions under the two-dimensional nonlinear Schrödinger equation with periodic boundary conditions.