An Approximate Spectral Method for Accurate Solution of Fourier and Non-Fourier Heat Conduction Problems

Abstract

A new form of the pseudo-spectral method is presented. The method is theoretically simple yet robust enough to produce very accurate solutions to hyperbolic 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 sub-domain 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