Development of a Grid Partial Analytical Solution Method for Solving the Moving Source Heat Conduction Problem

Abstract

The grid partial analytical solution method is a newly developed unconditionally stable explicit numerical solution method for solving parabolic partial differential equations. This method discretizes only the spatial domain and predicts a continuous time dependent function at each spatial nodal point. As such, instead of conventionally predicting the solution from a number set, this method predicts from a functional domain. The typical properties of the grid partial analytical solution method can be summarized as the following: (1) It predicts a continuous nodal time dependent function rather than a discrete nodal value. (2) The prediction is unconditionally stable. And unlike any other unconditionally stable finite difference schemes which will lose accuracy when Fourier number becomes large, the proposed method allows single step time marching and unlimited reduction in the spatial step size Δx. (3) For a fixed time step, the higher value of the grid Fourier number resulting from decreasing Δx, the higher the accuracy is achieved in the predicted solution. (4) The grid partial analytical solution converges uniformly to the full analytical solution as the spatial truncation error is infinitely decreased by reducing the spatial step size Δx. This unique characteristic of the analytical treatment of time also makes it possible to treat other time dependent nonhomogeneities involved in heat conduction problem analytically. In this paper, a moving source heat conduction problem is posed and its grid partial analytical solution method developed.