An Advanced Numerical Method for the Solution of Parabolic Differential Equations

Many methods have been suggested for the solution of parabolic partial differential equations. Previous work has focused on explicit difference equations, implicit difference equations, a combination of these two methods, and exponential methods. This paper introduces an advanced numerical method for developing solutions to parabolic partial differential equations. The method combines higher order mathematics with numerical methods to derive a solution algorithm.

An outline of the explicit, implicit, Crank-Nicolson, DuFort-Frankel, modified DuFort-Frankel, and exponential methods is presented. The advanced numerical method is then outlined and shown to be unconditionally stable for all time steps in the sample problems. Accuracy and speed of solution for this method is compared to other solution methods for various test cases. For the test cases solved, the advanced numerical method gave excellent results when compared to the analytical solution and to the results of other methods. The advanced numerical method yields superior speed of solution, accuracy, and stability in solving parabolic partial differential equation problems compared to the other methods evaluated.