Abstract
This dissertation investigates the numerical accuracy and computational efficiency of a
multi-domain higher-order compact finite difference method for solving time-dependent partial
differential equations. The study focuses on reducing computational time while maintaining
numerical accuracy. The proposed approach involves domain decomposition, dividing the
computational domain into smaller subdomains. Sixth-order compact finite difference schemes
are formulated to solve time-dependent partial differential equations. The quasilinearisation
technique is used to address nonlinearities in the equations by transforming nonlinear problems
into a sequence of linearised iteration schemes. These schemes are then solved using higherorder
compact finite difference methods and, in some cases, the Crank-Nicolson method, both
of which are well-known for their numerical stability and accuracy.
This study applies the multi-domain higher-order compact finite difference method to solve
reaction-diffusion equations, Burgers’ equations (including a coupled system of equations), and
Black-Scholes equation. The multi-domain approach is particularly useful for handling coupled
Burgers’ equations, where domain decomposition is used to modify the compact finite
difference schemes. The numerical results are validated against published solutions and, where
available, exact solutions. Findings from this research highlight that the multi-domain compact
finite difference method significantly improves computational efficiency while ensuring
high accuracy. The results demonstrate that this approach effectively controls numerical error,
making it a reliable method for solving complex partial differential equations.