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IBM Journal of Research and Development  
Volume 16, Number 3, Page 277 (1972)
Mechanical Technology
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Removal of Numerical Instability in the Solution of Nonlinear Heat Exchange Equations

by J. W. White
Current relaxation methods used by most large-scale thermal-computation programs for solving steady-state temperature distributions are subject to numerical instability when radiation is present in the system under study. The instability is usually due to the linearization of the numerical formulation of the governing equations and manifests itself as an oscillatory or divergent solution. A new method of solution is presented, which takes advantage of the diagonal dominance of the coefficient matrices of the linear and nonlinear transfer modes and uses a Gauss-Seidel system relaxation method together with a Newton-Raphson root-evaluation technique. System solution appears similar to that for a diagonally dominant linear system for all magnitudes of radiation. Test cases show the method to be monotonically convergent over the entire range of the radiation parameter considered, while previous methods were found to fail at certain magnitudes of the nonlinear term.
Related Subjects: Computation; Materials; Mathematics (applied); Mechanics and mechanisms; Radiative transfer; Thermodynamics