Mathematical Modelling of heat conduction-convection requires methods that are efficient and accurate. This is of paramount importance for modeling complex problems of value for the heat transfer community. In this paper, a One-dimensional Substantial Capacitance method with solutions free from oscillations is presented. The method is derived by equating energy and heat control volume equations. The method can be readily implemented into a standard finite element method (FEM) code. Resulting FEM equations are similar to those used to model classical heat-conduction. A 1-D semi-infinite bar is employed to perform numerical testing. Five methods are employed for code validation and these are analytical, Bubnov-Galerkin, Petrov-Galerkin, Substantial Capacitance and Finite Difference schemes (FDS); the approach is shown to be accurate, stable and computationally competitive.

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