Stiff equations Newton iteration

Fig. 15.7 Conceptual illustration of the behavior of a Newton iteration on a nonlinear, stiff system of algebraic equations. A contour map of a norm of the residual vector F is plotted. The curvature represents nonlinear behavior, and the elongation represents disparate scaling, or stiffness. The desired solution of the problem is represented by the X the current iteration is marked by a dot. The elliptical contours represent residuals of the local linearization at the current iterate.

We strongly suggest the use of the reduced sensitivity whenever we are dealing with differential equation models. Even if the system of differential equations is non-stiff at the optimum (when k=k ), when the parameters are far from their optimal values, the equations may become stiff temporarily for a few iterations of the Gauss-Newton method. Furthermore, since this transformation also results in better conditioning of the normal equations, we propose its use at all times. This transformation has been implemented in the program for ODE systems provided with this book. 

Often, the term stiff differential equation is used to indicate that special methods are used for numerically solving them. These methods are called stiff integrators and are characterized by A-stability or at least i4(a)-stability. They are always implicit and require a corrector iteration based on Newton s method. For example BDF methods or some implicit Runge-Kutta methods, like the Radau method are stiff integrators in that sense. 

We note that while the Newton method is the most robust and most widely used in nonlinear finite element software, it is also computationally expensive primarily due to the necessity to solve a system of linear equations. It also imposes considerable computer memory requirements since a global system matrix is used. This method also is not as easily parallelized as some other iterative methods. In order to achieve the optimal performance of the Newton method, it is crucial to calculate the tangent stiffness matrix that is indeed tangent or, in other words, is the derivative with respect to unknowns that are calculated very accurately. 

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