Quasi‐Newton iterative strategies applied to the heat diffusion equation

Abstract

SUMMARYSeveral iterative procedures have been used to solve the non-linear heat diffusion equation taking into account radiation across internal cavities. Quasi-Newton methods are compared with Picard iteration and Newton-Raphson methods. Among them, the rank-one quasi-Newton update seems to be the most effective, specially in time-dependent cases.

Finite-Element Modeling of Coupled Radiative and Diffusive Heat Transfer in Nonparticipating Media Including Symmetry and Periodicity Conditions

“…The method is similar to other proposed methods (see, for example, [6]), except that it involves a symmetric matrix. Moreover, Eq.…”

“…In practice, there is no method to ensure the convergence of the iterative process in all cases. Therefore the user has always the choice for the solution strategy of a given problem according to the nature of the problem to solve (see, for example, [6]). The full Newton± Raphson method, for example, consists of choosing for matrix [K] i the tangent matrix of the equations system:…”

“…Nevertheless, the solution of coupled radiative and di usive equations needs some special attention [2,5,6].…”

Solution strategies for coupled conduction/radiation problems

“…This type of approach was considered in [11,12] though no large-scale examples of use were demonstrated. For problems having a relatively small number of radiation surfaces, Equation (11) can be solved forq by inverting A and substituting the result into (10).…”

Modelling of Advection-Dominated Transport in a Porous Column with a Time-Decaying Flow Field

“…It has been noted that due to its efficiency and accuracy, the QuasiNewton Method (QNM) has been applied for the solution of the optimal problem in many areas in the engineering sciences (Engelman et al, 1981;Soria and Pegon, 1990;Gottlieb and DuChateau, 1996). Yu (1998) provides a mathematical proof of the efficiency of the QNM in relation to parameter identification associated with parabolic partial differential equations.…”