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ISSN Online: 2377-424X

ISBN Print: 0-89116-130-9

International Heat Transfer Conference 6
August, 7-11, 1978, Toronto, Canada

A Comparison of Theory and Experiments for Accuracy, Stability and Oscillations of Transient Heat Conduction

Get access (open in a dialog) DOI: 10.1615/IHTC6.360
pages 257-262

Sinopsis

Various numerical methods, in particular, finite element (FE), finite difference (FD) and weighted residuals methods (MWR, are reviewed. It was shown that the MWR-Collocation yields FD whereas the MWR-Galerkin yields FE and thus it may be concluded that the FE and FD belong to the class of MWR. Several Laplacian terms, including the most accurate 5-node and 9-node formulas as well as another new one (L1719) that unified FE, FD, and MWR; were examined by order of magnitude analysis to compare their accuracy. Even though, the new one is better than the familiar one (Ll8) of the same type, the finite element method can not be the best as far as accuracy is concerned. The stability and nonoscillation limits are given in the form of table for explicit, Crank-Nicholson type (Φ = 0.5), and standard implicit schemes. They do point out that the most accurate scheme doesn't necessarily possess the best stability and nonoscillation characteristics. It is also clear that the MWR-Galerkin requires smaller step sizes than the MWR-Collocation or finite differences to enforce stability or nonoscillations or both. The transient temperatures are predicted due to a step change in boundary conditions for a two-dimensional plate. The MWR-Galerkin/finite-element underestimates for small times and thus these may not be desirable for several time steps. The standard implicit scheme (Φ = l) overestimates for small times by about 50 percent and thus Crank-Nicholson type (Φ = .5) is preferred for initiation of computations at least for several time steps. The temperatures are lower for large times with standard implicit scheme. The optimization of Φ with respect to step sizes may very well improve the accuracy significantly. L17 is found to be less oscillatory and more stable and also more accurate than the other Laplacian approximations for explicit schemes. In general, the MWR-Collocation or finite differences is better than the MWR-Galerkin or finite element.