Backward implicit method

The backward implicit method consists of solving the equation... 

As anticipated, the equations of the system (6.7) are non-linear since they contain the product of two independent variables. An approximate way of solving the system is the linearisation of the non-linear terms. For example, in the case of (6.7) these correspond to the product of two unknown concentrations at the timestep k that, within the backward implicit method, can be approximated as [2]... 

Compton RG, PtUdngton MBG, Steam GM (1988) Mass transport in channel electrodes. The application of the backwards implicit method to electrode reactions (EC, ECE and DISP) involving coupled homogeneous kinetics. J Chem Soc Faraday Trans I 84 2155-2171... 

Stability analysis shows that the backward implicit method is stable for any choice of step size k and h. This means that the method is unconditionally stable as such, the stability does not depend on the a = kD/h value. 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

Implicit methods, which have far better stability properties than explicit methods, provide the computational approach to solving stiff problems. The simplest implicit method is the backward (implicit) Euler method, which is stated as... 

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... 

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... 

This implicit method uses a first-order backward difference approximation for the time derivative and a second-order central difference approximation for the spatial derivatives. The FDE is... 

Another possibility is to let the same derivative approximation pertain to the next time this is the backward implicit (BI) method ... 

It will be noted that the equation pair (11.10) contains further unknowns C jLi and C l , for i > 0. These can however be eliminated as described in Chap. 6, using the u-v mechanism. We assume that some implicit method is used here and that the first, backward, Thomas scan has been performed. Then, as described in that chapter, Sect. 6.2 or, for coupled systems, Sect. 6.4, concentrations can be expressed in the form... 

We may now note that this backward-difference formulation does not permit the explicit calculation of the Tp +1 in terms of Tp. Rather, a whole set of equations must be written for the entire nodal system and solved simultaneously to determine the temperatures Tp+. Thus we say that the backward-difference method produces an implicit formulation for the future temperatures in the transient analysis. The solution to the set of equations can be performed with the methods discussed in Chap. 3. 

The restriction on the step size (2.304) due to the stability condition for the explicit difference method can be avoided by using an implicit method. This means that (2.298) is discretised at time tk+1 and the backward difference quotient is used to replace the time derivative. With... 

If the integral on the right side of (12.55) is estimated using the value of the integrand at the final point, we obtain the first order implicit or backward Euler Method ... 

Implicit methods have been developed to overcome this problem. The backward Euler method is... 

Finite difference methods have been used bpth to test the assumptions made in the derivation of eqn. (27) under the Leveque approximation [35] and to solve electrochemical diffusion-kinetic problems with the full parabolic profile [36-38]. The suitability of the various finite difference methods commonly encountered has been thoroughly investigated by Anderson and Moldoveanu [37], who concluded that the backward implicit (BI) method is to be preferred to either the simple explicit method [39] or the Crank-Nichol-son implicit method [40]. 

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