An Approximate Spectral Method for Accurate Solution of Fourier and Non-Fourier Heat Conduction Problems(<Special Issue>Emerging Fields in Thermal Engineering)

In this paper a new form of the pseudo-spectral method is presented. The method is theoretically simple yet robust enough to produce very accurate solutions to hyper-bolic and parabolic PDE's while avoiding the effects of Gibb's phenomenon. Moreover the method uses a relatively small amount of computational memory. The method is based on the observation that an analytical function may be well represented in a set of small neighborhoods that share common boundaries, called sub-domains, by low order Chebyshev polynomials. A collocation solution scheme is used in each subdomain to march facilitate a time. Throughout this process the Chebyshev expansion coefficients of the highest order terms are monitored. If these coefficients grow beyond a specified small size, the new sub-domains are then redefined so that the function is again well represented by Chebyshev polynomial expansions. An approach for the determination of computational sub-domains of the physical domain for the special case of a discontinuous function is discussed. The strategy for solving the PDE's is presented. The method is then applied to Fourier and non-Fourier heat conduction problems.