The Explicit Method

The main goal of subsequent considerations is the comparison between ADM of the type (17) with parameters (25) and the explicit method with optimal set of Chebyshev s parameters... 

The method of steepest descent. The explicit method of steepest descent is given by the formulas... 

The explicit methods avoid the need of solving large sets of equations and can therefore be used on smaller computers. However, these methods tend to be unstable unless the step sizes are kept small and an artificial constraint is introduced on the variables. In most formulations the time step must be less than the reach length divided by the isothermal sonic velocity (see Section V,B,1). Explicit methods are used in PIPETRAN (D7) and SATAN (G4). 

The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations... 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

The main advantage of the implicit algorithms is that they do not become numerically unstable. Very large step sizes can be taken without having to worry about the instability problems that plague the explicit methods. Thus, the implicit methods are very useful for stiff systems. 

Figure 3-14 Schematics of dividing the diffusion medium into N equally spaced divisions. Starting from the initial condition (concentration at every nodes at f = 0), C of the interior node at the next time step (f = At) can be calculated using the explicit method, whereas C at the two ends can be obtained from the boundary condition.

Because the diffusion profiles at the next time step are calculated directly from initial and boundary conditions, this method is called the explicit method. The method is stable only when a <0.5, i.e., Ax/(A ) < 0.5, and has only first-order precision because the expression for (dCldx) has only first-order precision. Hence, given A, it is necessary to choose a small Ax. 

If D depends on concentration, the explicit method is easy to adapt but the implicit methods are more difficult. Let... 

The explicit method is implemented easily in a computer program or a spreadsheet. Unfortunately, there is an issue of stability that must be considered. It turns out that when the coefficient of wn-, the term in square brackets, becomes negative, the solution becomes unstable and the approach is completely unusable. It is evident by inspection that if the... 

Compared to the explicit Euler method (Eq. 15.9), note that the right-hand side is evaluated at the advanced time level tn+1- If f(t, ) is nonlinear then Eq. 15.22 must be solved iteratively to determine yn+. Despite this complication, the benefit of the implicit method lies in its excellent stability properties. The lower panel of Fig. 15.2 illustrates a graphical construction of the method. Note that the slope of the straight line between y +i and yn is tangent to the nearby solution at tn+, whereas in the explicit method (center panel) the slope is tangent to the nearby solution at t . 

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.

Because iteration is required for the implicit method, it takes more computational resources to solve the same problem using the same value of the time step. However, because of its significantly improved stability properties, the implicit method can take a much larger time step than the explicit method. Therefore there is the potential for the implicit method to be more efficient, as long as it can use larger time steps and still maintain sufficient accuracy. 

An improvement over the explicit method is done by the fully implicit method in which the concentration corresponding to time (j + 1) is calculated as... 

There are several methods that we can use to increase the order of approximation of the integral in eqn. (8.72). Two of the most common higher order explicit methods are the Adams-Bashforth (AB2) and the Runge-Kutta of second and fourth order. The Adams-Bashforth is a second order method that uses a combination of the past value of the function, as in the explicit method depicted in Fig. 8.19, and an average of the past two values, similar to the Crank-Nicholson method depicted in Fig. 8.21, and written as... 

Figure 8-2 Exact and numerical solutions obtained by the explicit method for the Cauchy problem (8-17) to (8-19), by using the spatial step h = 0.05 and the timestep At = 0.00125.

Both explicit and implicit methods are consistent and convergent. But, whereas the implicit method is absolutely stable, the explicit method is absolutely stable only if... 

As in the explicit method, the formulation of this implicit method is in two steps, as follows ... 

The truncation error for Eq. (10.30) is of the order (Ax) plus order (Ay)2. It is the same for Eq. (10.31). The explicit method shown by these equations are no longer used extensively because of the restrictive stability constraint. It is shown here for simplicity, and for discussion purposes. 

To have a better control on the stability of the explicit method by monitoring a single criterion, the second term in the x-momentum boundary layer equation can be depicted as... 

Problem Consider laminar flow of a fluid over a flat plate. Use the explicit method of finite differencing to compute the x-component velocity profile within the boundary layer. 

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. 

One of the main uses of digital simulation - for some workers, the only application - is for linear sweep (LSV) or cyclic voltammetry (CV). This is more demanding than simulation of step methods, for which the simulation usually spans one observation time unit, whereas in LSV or CV, the characteristic time r used to normalise time with is the time taken to sweep through one dimensionless potential unit (see Sect. 2.4.3) and typically, a sweep traverses around 24 of these units and a cyclic voltammogram twice that many. Thus, the explicit method is not very suitable, requiring rather many steps per unit, but will serve as a simple introduction. Also, the groundwork for the handling of boundary conditions for multispecies simulations is laid here. 

Kq exceeds the value 1000, the boundary conditions are taken to be those for a reversible reaction. How these two different boundary conditions are applied to calculate the concentrations C. yo and Cb,o is described below. Note that before new concentrations are to be computed, ail old concentrations, including the boundary values, must be known. When a new potential is stepped to, it comes into effect only after the concentrations are renewed, after which Co is calculated. This might be thought of as less than satisfactory, but it is consistent with the explicit method. In Chaps. 8 and 9, more satisfactory methods will be presented. 

In Chap. 5, the two-species cases were described for the explicit method. Here we add those for the implicit case. Both Dirichiet and derivative boundary conditions are of interest, the latter both with controlled current or quasire-versible and systems under controlled potential. 

Implicit methods have the great advantage of being stable, in contrast with the explicit method. It will be seen (and analysed in detail in Chap. 14) that the Laasonen method, a kind of BI, is very stable and responds to sharp transients with smoothly declining (but relatively large) errors, whereas Crank-Nicolson, also nominally stable, responds with error oscillations of declining amplitude, but is highly accurate. The drawbacks of both methods can be overcome, as will be described below. 

The Laasonen method, because of the forward difference in T, has errors of 0(6T, H2), and the first-order behaviour with respect to ST limits its accuracy to about the same as the explicit method described in Chap. 5. However, it has a smooth error response to disturbances such as an initial transient (Cottrell), and is stable for any value of 6T/H2, where // is either the same as all intervals if equal intervals are used in X, or is the smallest (usually the first) intervai if unequal intervals are used. This makes the method interesting, and it will be seen below that it can be improved. For simplicity, the symbol A will be used below, and denotes the largest value of that parameter, that is, the value from the smallest interval in space in a given system. 

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