Finite difference iterative schemes

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. 

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... 

These equations are now in convenient form for a finite-difference scheme along lines similar to that used above. Ail alternative approach developed in some detail employs the Lagrangian interpolation formula to follow the motion of the boundary in the (x/a, r) plane. This is a means of developing finite-difference approximations to derivatives based on functional values and not necessarily equally spaced in the argument. Crank points out that the application of Lagrangian interpolation formulas involves a relatively large number of steps in time, whereas the fixedboundary procedures require iterative solutions at each time interval, which are, however, far fewer in number. 

The momentum and energy equations are solved using a point-by-point iteration scheme. Derivatives are first replaced by finite differencies. A typical point is shown below... 

Thus the multiperiod optimisation problem is formulated as a sequence of two independent dynamic optimisation problems (PI and P2), with the total time minimised by a proper choice of the off cut variables in an outer problem (PO) and the quasi-steady state conditions appearing as a constraint in P2. The formulation is very similar to those presented by Mujtaba and Macchietto (1993) discussed in Chapter 5. For each iteration of PO, a complete solution of PI and P2 is required. Thus, even for an intermediate sub-optimal off cut recycle, a feasible quasi-steady state solution is calculated. The gradients of the objective function with respect to each decision variable (Rl or xRl) in problem PO were evaluated by a finite difference scheme (described in previous chapters) which again requires a complete solution of problem PI and P2 for each gradient evaluation (Mujtaba, 1989). 

For fixed or chosen values of the parameters, the model equations (eqs. 1-4) along with the initial and boundary conditions (eqs. 5) are solved iteratively by a centered-in-space, forward-in-time, finite difference scheme to obtain (i) the hexene and hexene oligomer concentration profiles in the pore fluid phase, and (ii) the coke (extractable + consolidated) accumulation profde. The effectiveness factor (rj) is estimated from the hexene concentration profile as follows ... 

In most practical cases the original relations (Eq. 16) are nonlinear and the linear least-squares treatment must be iterated to obtain convergence. The elements of the Jacobian X must be recalculated with each new iteration step. Although the least-squares procedure is said to be rather tolerant with respect to the precision of the Jacobian X, true derivatives should be used if ever possible, because finite difference schemes will most often require detailed considerations with respect to the allowed step width. Even then the results may show a tendency to oscillate long before a convergence limit due to the algorithms used or the number of digits carried is reached. With true derivatives, however, this limit is attainable. 

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

The most important feature of this dispersion-optimized FDTD method is the higher order nonstandard finite-difference schemes [6, 7] that substitute their conventional counterparts in the differentiation of Ampere s and Faraday s laws, as already described in (3.31). The proposed technique can be occasionally even 7 to 8 orders of magnitude more accurate than the fourth-order implementations of Chapter 2. Although the cost is slightly increased, the overall simulation benefits from the low resolutions and the reduced number of iterations. Thus, for spatial derivative approximation, the following two operators are defined ... 

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. 

The flow equation can be discretized by standard finite-difference schemes and then solved iteratively by the Newton-Raphson method. The detailed numerical procedure is similar to the work of Liu et al. [1] and will not be reported here. Once Q is determined, q can be computed using Eq. (7). Note that q represents the lateral flow uniformity. For a uniform liquid layer, q is equal to unity and then Q = y. 

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 ... 

The differential equations describing heat transfer through the fabric, air gap and skin are solved by a finite difference model. Due to the non-linear terms of absorption of incident radiation, Gauss-Seidel point-by-point iterative scheme is used to solve these equations. An under-relaxation procedure is utilized to avoid divergence of the iteration method. The Crank-Nicolson scheme [49] is used to solve the resulting ordinary differential equations in time. 

In order to correctly model the different possible states of the system, it will be necessary to cover a large part of the accessible phase space, so either trajectories must be very long or we must use many initial conditions. There are many ways to solve initial value problems such as (2.1) combined with an initial condition z(0) = 5. The methods introduced here all rely on the idea of a discretization with a finite stepsize h, and an iterative procedure that computes, starting from zo =, a sequence zi,Z2,..., where z z(nh). The simplest scheme is certainly Euler s method which advances the solution from timestep to timestep by the formula ... 

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