Novel numerical method for heat conduction using superposition of exact solutions

Abstract

A novel formulation for numerical solution of heat conduction problems using superposition of exact solutions (SES) to represent temperature on sub-elements of a region is described and demonstrated. A simple 1-D linear problem is used to describe the method and highlight potential benefits; however, extensions to higher geometric dimensions and linearization for temperature-dependent properties are possible. Significant new features of this method which are not part of conventional approaches are: local functions for temperature are both time- and space-dependent; both heat flux and temperature at the boundaries of each element have continuous representation; temperature and heat flux can be evaluated at every location in the domain; and heat fluxes on both internal and external boundaries are represented as linear functions of time. The SES method is validated against several exact solutions for the linear case (constant thermophysical properties) and is shown to have accuracy orders of magnitude greater than conventional Crank-Nicolson method. Extension to higher geometries and temperature-dependent properties are discussed.