Secant versus tangent methods in non‐linear heat transfer analysis

Abstract

The discretization procedure applied to the evolution equations of non-linear heat transfer is usually twofold; first, a semi-discretization in space is realized (by finite elements in the present case), leaving a set of first order non-linear differential equations and, second, the time integration of these equations is accomplished by a step-by-step procedure (e.g. by one-step implicit schemes), thus yielding a set of non-linear algebraic equations for each time-step considered. This paper discusses the two classes of methods one can resort to for solving these final non-linear algebraic systems: (1) Tangent methods derived from the Newton–Raphson (NR) or modified NR techniques, in which successive linearized systems based on the tangent Jacobian matrix are solved until coveragence is reached; owing to the second order accuracy of this method the required number of iterations per step is small, especially if the capacity heat flows are mildly non-linear; tangent methods applied to diffusion problems suffer, however, an important shortcoming; the Jacobian matrices are unsymmetrical and thus require extended core storage and special care in the organization of the solution procedure. Some economy may be realized by either the modified tangent method (the tangent matrix is kept constant for several iterates) or the restricted tangent method (a single iterate per time-step). (2) Secant methods obtained by direct linearization of the time-step equations, in which out-of-balance heat flows are computed at each iteration and serve to correct the solution until heat flow equilibrium is restored; in comparison with tangent methods, a greater number of iterations per step is expected for reasonable accuracy, but the system matrix is always built and inverted only once per time-step and, above all, is symmetrical. it is shown how the pseudo-force formulation with a few corrective loads updating enjoys nearly the same convergence properties as the modified tangent method without involving unsymmetrical system matrices. This constitutes the best suited method for mildly non-linear problems of heat transfer. Some simple numerical experiments are presented that show the relative advantages of the two methods on bench-mark problems.