Explicit second-order accurate taylor-galerkin-based finite-element formulations for linear/nonlinear transient heat transfer

This paper describes an explicit second-order accurate Taylor-Galerkin-based finite-element approach for transient linear/nonlinear heat transfer. The fundamental concepts and characteristics of the formulations and the associated solution methodology used are described in technical detail. The approach is based on expressing the finite-difference approximation of the transient time derivative in terms of a Taylor series expansion including higher-order time derivatives, which are then evaluated from the governing heat conduction equations. The resulting expressions are then discredited in space via the classical Galerkin scheme using finite-element formulations. Alternative formulations that employ the concept of flux representations are also developed to effectively enhance the discretized equations and to handle general nonlinear/linear boundary conditions. The stability and accuracy of the present formulations are also examined. Comparative results of several one- and two-dimensional test problems demonstrate the applicability of the proposed formulations for general transient linear/nonlinear heat transfer analysis.