A NEW NUMERICAL TECHNIQUE FOR THE SOLUTION TO THE HEAT EQUATION BASED ON A HYBRID PERTURBATION GALERKIN METHOD

摘要

A new technique for the numerical solution to the heat equation has been developed. A two-step hybrid analysis technique, which combines perturbation techniques with the Galerkin method, provides a systematic way to develop finite difference equations. The hybrid method is applied to heat transfer problems that can be characterized as: 1) steady state, one-dimensional, convective transport; 2) non-steady, one-dimensional, convective transport; and 3) non-steady, heat conduction with variable thermal conductivity. The main findings include: 1) perturbation terms result in the classical central difference equations; 2) local truncation error terms of the hybrid difference equations show improved finite difference schemes, i.e., smaller error terms relative to the classical central difference equations and that larger grid spacing may be chosen; 3) explicit, fully implicit and Crank-Nicolson type difference formulations are developed; and 4) for a given grid spacing, a stability analysis indicates that larger time steps may be selected relative to that found in the classical explicit method for mesh Reynolds number R > 2.34.