Abstract:
The finite difference approximation is a numerical method for solving differe-
tial equations. The basic idea for the solution is to approximate a differential
equation by a system of algebraic equations and is to replace the derivatives in
the equation by finite difference. We use a programming language, for example,
MATLAB, to solve resulting systems numerically. We present in this thesis the
finite difference approximation for solving parabolic differential equations in
one dimension. We discuss an extension of the finite difference approximation
to solve parabolic systems in higher dimensions. We present some theorems for
the convergence of the numerical approximation and we analyze some schemes
for their stability and convergence. We consider finite difference schemes in two
spatial dimensions. One difficulty associated with schemes in more than one
dimensionis that the Van Neumann Stability analysis can become formidable.
We also introduce the alternating direction implicity method (ADI) which is
among the most useful of the methods for multidimensional problems.
