Implicit solutions

If 3 u/3x is represented by a central difference expression and du/dy by a backward difference expression an implicit solution may be obtained where... 

A general, albeit implicit solution for arbitrary field dependence of p has beer presented by Young [38[. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

An approximate implicit solution (van Krevelen and Hoftijzer, 1948) for the enhancement... 

The problem of Example 7.3 will again be solved with explicit and implicit exponential differences, and compared with the analytical solution, equation (E7.4.7). This solution is given in Figure E7.5.1. Note that the explicit solution is close to the analytical solution, but at a Courant number of 0.5, whereas the implicit solution could solve the problem with less accuracy at a Courant number of 5. In addition, the diffusion number of the explicit solution was 0.4, below the limit of Di < 0.5. The implicit solution does not need to meet this criteria and had Di = 4. 

Figure E7.5.1. Comparison of explicit, implicit, and analytical solutions for the filter problem. Fe = 1.25 for explicit and implicit solutions. Cou, Courant number.

As stated above, the spatial derivative is approximated without regard to the time level. The distinction between explicit and implicit solutions depends on the time level at which the spatial derivatives are evaluated. Finite-difference stencils for explicit and implicit Euler methods are illustrated in Fig. 4.13. 

There are a variety of possible solution algorithms, which may be catatorized broadly as either explicit or implicit [13], The lower two panels of Fig. 15.2 illustrate graphically the construction of the most-straightforward explicit and implicit solution algorithms. 

As discussed in Section 15.3.2 on the implicit solution of transient differential equations, one step of the backward Euler method takes the form... 

In this section we describe the spreadsheets used to solve the Stokes problem between a cylindrical shell and an inner rod that rotates with fixed rotation rate, Section 4.8. Both explicit and implicit solution procedures are illustrated. This problem has boundary conditions that are fixed in time, and solves the transient problem to the steady-state solution. Other problems discussed in Chapter 4 have time-varying boundary conditions or time-varying forcing functions. Solving these problems requires only very straightforward modification of the following examples. 

Figure D.4 illustrates a spreadsheet that implements an implicit solution to the problem described in Section 4.8. The differences in the spreadsheet for the implicit method and the explicit method in the previous section begin in cell D21, where the difference formula is entered.

Solve the second equation for 5, which produces 8 = p (X X) 1X y = p b. Insert this implicit solution into the first equation to produce n/p = Z, y (py - px/b). By taking p outside the summation and multiplying the entire expression by p, we obtain n = p2 Z, y, (y, - x/b) or p2 = n/[Z y (y - x/b)]. This is an analytic solution for p that is only in terms of the data - b is a sample statistic. Inserting the square root of this result into the solution for 8 produces the second result we need. By pursuing this a bit further, you canshow that the solution for p2 is just n/e e from the original least squares regression, and the solution for 8 is just b times this solution for p. The second derivatives matrix is... 

Extend Figure 8.3 to the higher values of a t / R2 needed to show an asymptotic approach to the performance of a CSTR. Assume LjR = 16. A partially implicit solution technique is suggested. See Appendix 8.3. 

Implicit solution of the spreading pressure integral ci° = f(jimix, c °)... 

We observe that a maximum exists for the concentration of species B. Sometimes, Maple gives implicit solutions, i.e., independent variable (t), as a function of the dependent variable (y). 

For implicite solutions and steady state problems the initial guess of the unknown fields are given physical values, as close to the expected solution as... 

For implicit solution methods other approximate and rather crude outlet conditions are sometimes used for the flow variables. In the commercial flow... 

The solution of this equation involves an integral on the left-hand side that results in an implicit solution for h[ t ]. We can try to solve directly for h[ t ], but Mathematica cannot do it ... 

Two principal features characterize the systematic approach to equilibrium calculations used in this book a) Expressing concentrations of every species by the product of a (a fraction of that species of all others in the same system. These fractions are a function of only the critical variable (e.g., pH), the relevant equilibrium constants, and C, the total concentration of the component, and b) Describing the equilibrium condition by a single balance equation, e.g., the proton balance equation (PBE), the ligand balance equation, etc. This results ultimately in a description of the equilibrium condition of the solution by one equation with a single concentration variable, i.e., in an implicit solution. 

Fig. 9.5. Z-implicit and il-implicit solution sweeps. In the current sweep, thin-lined columns are already solved for, and the thick-lined columns are those currently being solved.

Now perform a structural analysis to determine the best set of design variables. You can use a modified functionality matrix to make this analysis easier. Enter an A if the variable can be uniquely solved (linear, exponential, etc.) and a B if a multiple root exists or implicit solution is needed. 

With the discrete pressure equation written for each fluid column, and by Including the pressure boundary conditions, the (imax jmax) unknown discrete pressures can be expressed In terms of (loiax jBiax) equations. This work solved the system of equations by using a modified strongly Implicit solution procedure (HSIP) [23], and established a technique Whereby the cavitation boundary could be located at any position between adjacent pressure grid points. 

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

In this chapter, an implicit solution for the equilibrium thickness of a mudcake was given assuming that its yield shear stress was known and that the fluid was laminar and Newtonian. Extend the solution to (i) turbulent Newtonian, (ii) laminar power law, and (iii) turbulent power law fluids. 

Chen, K. S. and M. A. Hickner. Modeling PEM fuel cell performance using the finite-element method and a fuUy-coupled implicit solution scheme via Newton s technique. Proceedings of the 4th International Conference on Fuel Cell Science, Engineering and Technology, Irvine, California, July 19-21,2006. 

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. 

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