Defense Date


Document Type


Degree Name

Master of Science


Mathematical Sciences

First Advisor

Jerrold S. Rosenbaum


In order to obtain a numerical solution to the heat equation using finite differences, either implicit or explicit equations are used to formulate a solution. The advantage in an explicit formulation is its simplicity and minimal computer storage requirements while its disadvantage is its instability. The opposite is true for an implicit formulation such as the Crank-Nicolson method; although it is stable it is more difficult to implement and requires a much larger memory capacity. In this paper we examine the accuracy and stability of a hybrid approach, a modified Crank-Nicolson formulation, that combines the advantageous features of both the implicit and explicit formulations. This hybrid approach results in a 20% reduction in the amount of work required compared to the standard Crank-Nicolson solution if both methods use a special tridiagonal system solver. If Gaussian elimination is used, the modified Crank-Nicolson approach reduces the amount of work by 87%. Regardless of the linear system solver used, the modified Crank-Nicolson approach reduces by 50% the memory requirement of the standard Crank-Nicolson method.


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