Nonlinear terms Newton iteration

Steady-state solutions are found by iterative solution of the nonlinear residual equations R(a,P) = 0 using Newton s methods, as described elsewhere (28). Contributions to the Jacobian matrix are formed explicitly in terms of the finite element coefficients for the interface shape and the field variables. Special matrix software (31) is used for Gaussian elimination of the linear equation sets which result at each Newton iteration. This software accounts for the special "arrow structure of the Jacobian matrix and computes an LU-decomposition of the matrix so that qu2usi-Newton iteration schemes can be used for additional savings. 

The modified Newton iteration, and the reason that damping is effective, can be explained in physical terms. Chemical-kinetics problems often have an enormous range of characteristic scales—this is the source of stiffness, as discussed earlier. These problems are also highly nonlinear. 

The resulting C + E equations are nonlinear in unknowns nj, nj, and tt but In nj are iteration variables since nj occur in logarithmic terms. These equations are linearized using first-order Taylor Series (Newton-Raphson method), in the variables An , A (In nj), and ir, and with n nj are reduced to S + 1 + E linear equations in unknowns AN, A (In N), and tt. When extended to include P mixed phases, we nave shown that they are nearly identical to the equations of the RAND Method and have the same coefficient matrix. 

An alternative, called semi-implicit methods in such texts as [2], avoids the problems, and some of the variants are L-stable (see Chap. 15 for an explanation of this term), a desirable property. This was devised by Rosenbrock in 1962 [35]. There are two strong points about this set of formulae. One is that the constants in the implicit set of equations for the k s are chosen such that each ki can be evaluated explicitly by easy rearrangement of each equation. The other is that the method lends itself ideally to nonlinear functions, not requiring iteration (as with the Newton method), because it is, in a sense, already built-in. This is explained below. 

Due to the nonlinearity of the discretized convective terms, resp., of the reinitialization step, iterative defect correction or Newton-like methods, resp., corrections via redistancing, must be invoked in steps (1) and (4). However, due to the assumed relatively small time steps, such nonlinear iteration methods are not critical for the complete flow simulation. 

In these expressions, the cluster operator is constructed from the amplitudes t ". In Newton s method, an iterative scheme is established by setting the expanded vector function equal to zero and neglecting terms that are nonlinear in At ... 

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