Secant versus tangent methods in non-linear heat transfer analysis

In recent nonlinear finite element applications, much interest has been focused on the solution of nonlinear field problems governed by a parabolic differential equation; of special concern is the time integration of the nonlinear systems of first-order ordinary differential equations resulting from a spatial discretization of the unknown by finite elements. This paper discusses and compares two methods for solving this initial value problem, taking as a physical example the heat transfer problem: tangent methods derived from the Newton-Raphson technique; and secant methods obtained by direct linearization of the time-step equations. Some simple numerical experiments are presented that show the relative advantages of the two methods on bench-mark problems.