Predicting Oscillatory Finite Difference Solutions to the Heat Equation: A Comparative Study of the Coefficient and Matrix Methods

摘要

This thesis compared the coefficient method with the full matrix method for predicting stability and oscillatory behavior of Finite Difference Methods, FDMS, used in solving the one and two dimensional transient heat (diffusion) equation with Dirichlet boundary conditions. Five FDMs were used: the fully implicit, fully explicit, DuFort-Frankel, Crank-Nicolson, and Peaceman-Rachford ADI. Analytically, the Pade' method was found to be equivalent to the matrix method in predicting stability and oscillations. The matrix method was shown to be more severe than the coefficient method in predicting both stable and non-oscillatory step size constraints. Also, the matrix method, while mathematically more rigorous, proved to be more difficult to derive and analyze, possibly limiting its usefulness. Ultimately, it was found that the coefficient method's predictions could be derived from the matrix method for two-level FDMs. Numerically, all methods were solved repeatedly and the results investigated for oscillatory behavior and maximum errors in the h-k, or space and time step, domain.