Difference scheme implicit

A much more pronounced vortex formation in expanding combustion products was found by Rosenblatt and Hassig (1986), who employed the DICE code to simulate deflagrative combustion of a large, cylindrical, natural gas-air cloud. DICE is a Eulerian code which solves the dynamic equations of motion using an implicit difference scheme. Its principles are analogous to the ICE code described by Harlow and Amsden (1971). 

For large values of z a fully developed case is reached in which the velocities are only functions of r and 0. In the fully developed case the weight fraction polymer increases linearly in z with the same slope for all r and 0. An implicit finite difference scheme was used to solve the model equations, and for the fully developed case the finite difference method was combined with a continuation method in order to efficiently obtain solutions as a function of the parameters (see Reference 14). It was determined that except for very large Grashof... 

Show that for any r and h a pure implicit difference scheme (a forward difference scheme) approximating the problem... 

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. 

Other ideas are connected with two types of purely implicit difference schemes (the forward ones with cr = 1) available for the simplest quasi-linear heat conduction equation... 

The scheme ascribed to Peaceman and Rachford provides some realization of this idea and refers to implicit alternating direction schemes. The present values y = and y = of this difference scheme are put... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

Latrobe, A., 1978, A Comparison of Some Implicit Finite Difference Schemes Used in Flow Boiling Analysis, in Transient Two-Phase Flow Proc. 2nd Specialists Meeting, OECD Comm, for Safety of Nuclear Installations, Paris, Vol. 1,439-495. (3)... 

Stiimer and House (1989) suggested an inverse relationship between the levels of chemical input and the system sustainability, and their principle is widely, more or less implicitly, accepted Zandstra (1994, as reported by Hansen 1996), however, proposed a different scheme, with insufficient chemical inputs leading to exhaustion of natural resources and excessive inputs leading to accumulation and eventually to pollution. 

Explicit different schemes show poor stability properties (Mitchell, 1969). In terms of the central difference operator, it may be shown that an accurate implicit equation is... 

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. 

The variational principle has not been widely used in diffusion kinetic problems. Nevertheless, it is such a powerful technique that it is suitable for discussing the many-body problems which have still to be tackled. Wherever approximate methods are necessary, the variational principle should be considered. The trial function(s) should be chosen with care, based on a good idea of the nature of the trial function from its behaviour in certain asymptotic limits. The only application known to the author of the variation principle to a numerical study of a diffusion kinetic problem on a molecular system is that of Delair et al. [377]. They used the variational principle to generate an implicit finite difference scheme for solving the Debye—Smoluchowski equation. Interesting comments have been made by Brykalski and Krason more in the context of heat diffusion [510]. 

The scheme ascribed to Peaceman and Rachford provides some realization of this idea and refers to implicit alternating direction schemes. The present values y = yn and y = yn+1 of this difference scheme are put together with the intermediate value y = j/n+1 2, a formal treatment of which is the value of y at moment t = tn+]/2 = tn + r/ 2. The passage from the nth layer to the (n + 1)th layer can be done in two steps with the appropriate spacings 0.5 r ... 

To illustrate the usage of implicit finite difference schemes, we will solve the cooling process of an amorphous polymer plate. Since an amorphous polymer does not go through a crystallization process we will assume a constant specific heat2. The implicit finite difference for this equation can be written as,... 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

The governing equations of the model are discretized in space by means of the finite element method [3, 18], and in time through a fully implicit finite difference scheme (backward difference) [18], resulting in the nonlinear equation set of the following form, [4, 7],... 

This is the so-called Crank-Nicholson scheme and, formally, it could have been obtained by simply averaging the explicit forward-difference and implicit backward-difference schemes. By conveniently grouping the terms, the following system of linear equations results ... 

The equation for the central point (i = 1) actually plays the role of inner boundary condition. The above system should be completed with one more boundary condition for the outer point tm = R. Irrespective of the type of the used time difference scheme (explicit, fully implicit or Crank-Nicholson), the further treatment of the resulting system of difference equations is absolutely analogous to the one developed for Cartesian coordinates. 

It is observed that the above finite difference scheme is implicit if p> Vi. The finite difference equation (10.41) may be used as the continuity equation for both the fiilly implicit and the explicit methods. 

The finite difference approximations can either be applied to the derivatives on the line from which the solution is advancing or on the line to which it is advancing, the former giving an explicit finite difference scheme and the latter an implicit scheme. The type of solution procedure obtained with the two schemes is illustrated in Fig. 3.18. 

The contaminant transport model, Eq. (28), was solved using the backwards in time alternating direction implicit (ADI) finite difference scheme subject to a zero dispersive flux boundary condition applied to all outer boundaries of the numerical domain with the exception of the NAPL-water interface where concentrations were kept constant at the 1,1,2-TCA solubility limit Cs. The ground-water model, Eq. (31), was solved using an implicit finite difference scheme subject to constant head boundaries on the left and right of the numerical domain, and no-flux boundary conditions for the top and bottom boundaries, corresponding to the confining layer and impermeable bedrock, respectively, as... 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

S. S. Abarbanel and A. E. Chertock. Strict stability of high-order compact implicit finite-difference schemes the role of boundary conditions for hyperbolic PDEs, I. J. Comput. Phys., 160 42-66, 2000. 

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... 

However, there is a better choice. If we believe the limit function to be Ck, then we choose as our sequence of functions the B-splines of degree k +1. This is actually the assumption which has been made implicitly in the use of difference schemes above. If, for example, a scheme has a convergent second difference scheme, then we can think of this second difference scheme as having a polygon, so that the second derivative varies linearly. If the second derivatives vary linearly in each span, then the first derivatives vary quadrat-ically and the actual values cubically, and so, because the support is finite, the implicit approximant is a sequence of cubic B-splines28. 

In this section, an explicit time advance scheme for unsteady flow problems is outlined [30]. The momentum equation is discretized by an explicit scheme, and a Poisson equation is solved for the pressure to enforce continuity. The continuity is discretized in an implicit manner. In the original formulation, the spatial derivatives were approximated by finite difference schemes. 

The additive scheme implicitly assumes that the contributions of the different substructures to the molecule activity are statistically independent of each other. 

Coefficients a, a2 and b are obtained by the Fourier analysis and the relatively rapid solution of the resulting tridiagonal system of equations, due to the implicit nature of (2.31). A typical set is a = 22, a% = 1, and b = 24. To comprehend their function, let us observe Figure 2.2 that assumes the computation of dHy/dx and dEz/dx at i = 0. For the first case, constraint Ey = Ez = Hx = 0 at i = 0 indicates that dHy/dx (likewise for all H derivatives) must also be zero. In the second case, to calculate dEz/dx at i = one needs its values at i = —, . Nonetheless, point i = — is outside the domain and to find a reliable value for the tridiagonal matrix, the explicit, sixth-order central-difference scheme is selected... 

---