Higher order parallel splitting methods for parabolic partial differential equations

Abstract

The thesis develops two families of numerical methods, based upon new rational approximations to the matrix exponential function, for solving second-order parabolic partial differential equations. These methods are L-stable, third- and fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods second-order spatial derivatives are approximated by new difference approximations. Then parallel algorithms are developed and tested on one-, two- and three-dimensional heat equations, with constant coefficients, subject to homogeneous boundary conditions with discontinuities between initial and boundary conditions. The schemes are seen to have high accuracy.

A family of cubic polynomials, with a natural number dependent coefficients, is also introduced. Each member of this family has real zeros.

Third- and fourth-order methods are also developed for one-dimensional heat equation subject to time-dependent boundary conditions, approximating the integral term in a new way, and tested on a variety of problems from the literature.