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Newton-Iteration Method

  We can linearize the system by a Taylor series expansion as given by (4.1-23), where tex2html_wrap_inline4939 gives the Jacobian matrix tex2html_wrap_inline4941 and tex2html_wrap_inline4943 denotes the update vector.

  equation624

For the Newton iteration method the higher order terms of (4.1-23) are neglected and the linearized equation system (4.1-24) at the tex2html_wrap_inline4945 iteration is solved instead.

  equation638

tex2html_wrap_inline4947 is referred to as right-hand-side (RHS) vector. To solve the nonlinear system we have to start the Newton iteration with an initial guess tex2html_wrap_inline4949 for the solution variables. This initial guess is coming from the initial conditions at the first time step or taken from the previous time step. The new solution of the equation reads:

  equation650

In the case of a linear equation system, tex2html_wrap_inline4951 would be the exact solution. As diffusion systems are nonlinear the Newton method exhibits a tendency to overestimate the update tex2html_wrap_inline4953 . This leads to intermediate results which can lie outside the convergence region of the Newton scheme or causes numerical problems which totally prevent convergence. Hence a damping factor d;SPMlt;1 is introduced to avoid this overshoot. Only a fraction of the update vector is added to the tex2html_wrap_inline4945 iterate solution vector. Several successfully applied damping algorithms can be found in the literature [Ban81] [Ban80] [Deu74].

The iteration scheme is terminated when the norm of the update vector tex2html_wrap_inline4959 is smaller than a specified Newton accuracy tex2html_wrap_inline4961 .

  equation666

Additionally, the residual of the nonlinear system is checked to be smaller than a chosen absolute accuracy tex2html_wrap_inline4963 .

  equation671



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