Difference scheme implicit iteration

The nonlinear part of the susceptibility was introduced into the quasi-linear finite-difference scheme via iterations, so that at any longitudinal point, the magnitude of E calculated at the previous longitudinal point was used as a zero approximation. This approach is better than the split-step method since it allows one to jointly simulate both the mode field diffraction on irregular sections of the waveguide and the self-action effect by introducing the nonlinear permittivity into the implicit finite-difference scheme which describes the propagation of the total field. 

In this chapter the new difference schemes are constructed for the quasilin-ear heat conduction equation and equations of gas dynamics with placing a special emphasis on iterative methods available for solving nonlinear difference equations. Among other things, the convergence of Newton s method is established for implicit schemes of gas dynamics. 

Fine wall mesh schemes have been used to avoid this patching process. It is critical to use a good implicit-difference scheme in this case. Mellor (Ml) developed a good linearized iteration technique which has since been adopted by others. 

Most published computations have dealt with boundary layers. The numerical techniques employed have varied considerably, and hence the computational costs initially varied widely among programs. But now most workers have adopted implicit-difference schemes, with special wall-region treatment as outlined above, and/or a linearized iteration technique (Ml), so that run times are now reasonably uniform. A typical two-dimensional compressible boundary layer can now be treated in under one minute on a typical large computer. 

Using the implicit difference scheme with respect to time in equations for u and v, and non-implicit difference scheme with respect to time in equation for h the system (4.6) was solved. The equation (4.8) was solved by implicit difference scheme with respect to x. To solve the system of non-linear equations appearing after discretization of governing equations, the iteration procedures were implemented. To avoid the singularity caused by condition h=0 far Irom the flow region, this condition was replaced by condition h=h >0. The value of h was chosen in a way that the solution in the rivulet area was independent from this parameter. In calculations the value of h was equal to... 

The solution of problem was carried out by the numerical finite-difference method briefly presented in [1], The calculating domain was covered by uniform grid with 76x46 mesh points. A presence of two-order elliptic operators in all equations of the mathematical model allow us to approximate each equation by implicit iterative finite-difference splitting-up scheme with stabilizing correction. The scheme in general form looks as follows ... 

Dewey et al. (D3) present a numerical scheme for the ablation of an annulus with specified heat fluxes at the outer (ablating) surface and at the inner surface. An implicit finite difference technique is used which permits arbitrary variation of the surface conditions with time, and which allows iterative matching of either heat flux or temperature with external chemical kinetics. The initial temperature may also be an arbitrary function of radial distance. The moving boundary is eliminated by a transformation similar to Eq. (80). In addition a new dependent variable is introduced to... 

The equation is solved by means of iteration and fully-implicit finite difference in the following scheme (Wen, 1997b),... 

Mass balance of solid Mass balance of water Mass balance of air Momentum balance for the medium Internal energy balance for the medium The resulting system of Partial Differential Equations is solved numerically. Finite element method is used for the spatial discretization while finite differences are used for the temporal discretization. The discretization in time is linear and the implicit scheme uses two intermediate points, t and t between the initial 1 and final t limes. Finally, since the problems are nonlinear, the Newton-Raphson method has been adopted following an iterative scheme. 

Table 5.1 Comparison of performance results for the implicit Euler scheme applied to different formulations of the pendulum problem. NFE number of function evaluations, NJC number of Jacobian evaluations, NIT average number of Newton iterations, ep,ey and ey. absolute errors in pi(4),ui(4), A(4), resp.

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