First-order implicit scheme

This paper describes a feasibility and prototyping study of multi-step reaction sequence representation and searching, using REACCS as a starting point. The results of this study will show the benefits and costs of a few alternatives, and the utility of such a system in the hands of practising synthetic chemists. Detailed studies were carried out on probable architectures of systems which support Explicit- and First-Order Implicit schemes options for the support of Second-order Implicit schemes are described in theoretical terms only. 

Figure 10. Comparison of search times and number of hits between standard REACCS RSS (equivalent to automatic prior registration), search-time discovery of explicit schemes, and first-order implicit schemes...

A Scheme-RSS search over both explicit and first-order implicit schemes, using the method of search-time discovery. The search was carried out for one-, two-, and three-step schemes. 

Although Fugmann and others hinted that the searching of first-order implicit schemes would return a wealth of reaction information, we found the results in many cases to be rather sparse, especially with consideration of the two- to 50-fold increase in computer resources necessary to find them. On the other hand, the added time necessary to complete a search over implicit schemes could be well worth the investment if the number of answers could be increased. Methods for accomplishing this are described in the next section. 

One can think of approaches to second-order implicit schemes in terms of how this ratio can be increased in a database. One approach is to reduce artificially the number of molecules in the database another is to increase the number of reactions, while holding the number of molecules constant. There is an important difference between approaches to second-order implicit sequences, and the t3q>es of schemes described previously. ExpUcit and first-order implicit schemes consist entirely of recorded fact, and are guaranteed to work in the laboratory. In order to rebalance the equation to return more answers to scheme queries, some abstract or empirical chemical knowledge must be used the process reliability is therefore not guaranteed, and the quality of results obtained will be based largely on the viabihty and generality of the rules by which the balance of the equation is modified. 

We found that the number of answers to reaction substructure queries over first-order implicit schemes was disappointing in general, compared to the amount of computer resources required to execute the search. Although many chemists will be willing to wait for those results, the authors are not conviced that a majority of users would consider the added results to be worth the decrease in performance. 

More basically, LB with its collision rules is intrinsically simpler than most FV schemes, since the LB equation is a fully explicit first-order discretized scheme (though second-order accurate in space and time), while temporal discretization in FV often exploits the Crank-Nicolson or some other mixed (i.e., implicit) scheme (see, e.g., Patankar, 1980) and the numerical accuracy in FV provided by first-order approximations is usually insufficient (Abbott and Basco, 1989). Note that fully explicit means that the value of any variable at a particular moment in time is calculated from the values of variables at the previous moment in time only this calculation is much simpler than that with any other implicit scheme. 

To discretize the continuity, the differential equation is first integrated over the scalar grid cell volume. This is the same grid cell volume as was used for the generalized scalar quantity in Sect. 12.8. By use of the first order implicit Euler time discretization scheme and the basic theorem of integration for the convective fluxes, a discrete form of (12.195) can be expressed as ... 

Finite Volume Methods The finite volume method, when the permeabihty tensor is diagonal in the selected coordinate system, approximates the pressure and saturation functions as piecewise constant in each grid block. The flux components are assumed constant in their related half-cells. Thus when two cells are joined by a face, the related component of flux is assumed to be the same each side of the face. The balance laws are invoked separately on each grid block, and are discretised in time either by an explicit or fully implicit first order Euler scheme or other variant as discussed in the previous subsection. 

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain... 

The inclusion of a primary radical termination process in radical polymerization schema usually leads to kinetic equations which cannot be reduced to a straightforward expression for the orders of reaction with respect to initiator and monomer concentration (48). It is interesting to note therdbre that, using only the normal approximations, such an expt ion can be derived from the above scheme which predicts that the polymerization will be half-or r in initiator and light intensity and first-order in monomer concentration, despite the participation of primary radical termination. Straightforward solution of the kinetics is made possiUe by the following assumptions, implicit in the scheme ... 

The first order explicit upwind scheme was introduced by Courant, Isaacson and Reeves [31], and later on several extensions to second order accuracy and implicit time integrations have been developed. 

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

In order to allow longer time steps to be used, implicit schemes must be used for the C, D and S operators in the first step of the momentum equation solution. 

When the first-order scheme is semi-implicit with 0 < a < 1, this realizability condition is a mix of those found for pure advection and pure diffusion. 

Writing a computer program for an implicit scheme requires some matrix manipulations and is somewhat involved. It is also first-order accurate in time but unconditionally stable. 

For numerical calculation, consistency of units and order of variables are important. Therefore, a system of equations (1) - (12) is converted to a dimensionless form. To solve these equations, an explicit/implicit scheme is used [10]. First of all, fluid properties and physical parameters of reservoir are set. Further calculations are conducted in the following order ... 

By this means, the partial differential equation is transferred into an ordinary differential equation in the discrete cosine space. A semi-implicit method is used to trade-off the stability, computing time, and accuracy [39,40]. In order to remove the shortcomings with the small time-step size associated with the exphcit Euler scheme to achieve convergence, the linear fourth-order operators can be treated implicitly while the nonlinear terms can be treated explicitly. The resulting first-order semi-implicit Fourier scheme is ... 

We can further subdivide implicit scheme searches into two categories. If the eocact product of one reaction is used as the reactant in another reaction, then the connection between the two reactions is by exact-structure, and is referred to here as a first-order imphcit sequence. If the connection between the two reactions is by inexact structure match (such as two highly similar molecules), then we refer to it as a second-order implicit sequence. This latter type of search has much in common with synthesis planning programs it has the most to offer, but poses the greatest technical challenge. 

If one removes the restriction that steps in a scheme be contained within a single document, a computer program can be used to find schemes which are contained implicitly in a database. If the product of a preceding step is used directly as a reactant in the following step, we refer to it as the first-order type. The reactions may never have been carried out as a sequence in the laboratory nonetheless, these sequences are always viable, although not necessarily valuable or interesting. 

For the solution of the transient problem, it is necessary to have a robust timestepping algorithm. The matrix equations result in a first-order equation in time. For the solution of this problem a three-level unconditionally stable scheme has been proposed [70]. One describes the three-level scheme class of time-step algorithm in the two-level scheme, which can vary between explicit, and implicit solution strategies [67]. The explicit scheme requires no matrix inversion, but the time step is limited by stability consideration. On the other hand, the implicit method is unconditionally stable but involves matrix inversion. 

Classification of Simulation Methods by Time Stepping Scheme Commercial flow simulators generally discretise time derivatives using a first order finite difference formula (Euler s method). The time derivative thus involves the difference of functions at the end and at the start of each time step. All other terms in the equations are discretised to involve functions evaluated at the start and the end of each time step. The pressure always appears at the end of the time step and one says that the pressure is implicit. Saturations appear at the end of the time step in the fully implicit approach. The saturation... 

The RELAP5/MOD3 code is based on a non-homogeneous and non-equilibrium model for the two-phase system that is solved by a fast, partially implicit numerical scheme to permit economical calculation of system transients. The objective of the RELAP5 development effort from the outset was to produce a code that included important first-order effects necessary for accurate prediction of system transients but that was sufficiently simple and cost effective so that parametric or sensitivity studies were possible. 

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