Backward implicit

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... 

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

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]. 

The temperature field is solved using a backward implicit difference approximation. The nodal temperature and variable rate of temperature of element are given by the interpolation of shape function. In the solution of temperature field, the equation of heat transfer, initial conditions and boundary conditions have to be satisfied. Based on the variational principle, the problem can be converted into solving the extremum of a functional. The implicit difference equation is written as ... 

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

Time integration methods more advanced than the one considered in this book (backward implicit, BI) can also be employed. Indeed, the Crank-Nicholson and high-order extrapolation methods [6] have proven to enable the reduction of the number of timesteps (and even improve the accuracy of the simulation) with respect to BI [4]. 

By approximating the spatial derivatives according to the central three-point difference formula and using the backward implicit scheme, the form of the resulting equations for species A and B are analogous to those discussed in previous chapters ... 

However, for fast reactions the AT value must be small enough such that the chemical change in the timestep is also small and the finite difference approximation is accurate. Note that this means a serious limitation if explicit methods are used for simulations given the restrictions in the AT/AX" value discussed in Chapter 3. When unconditionally stable methods are employed, like the backward implicit approach followed in this book, this problem is overcome such that one can alter the temporal and spatial grids independently, which makes the simulation much more efficient. 

After normalisation and discretisation of the problem with the central three-point approximation for the spatial derivative and the backward implicit scheme the differential equations of a point i in solution can be written in the form... 

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]... 

The resolution of the problem with the backward implicit scheme can also be carried out in a simple way given that the mathematical problem is formally equivalent to the diffusion-only one studied in Chapters 2 4. Thus, the convection-diffusion differential equation in discrete form for species j at the point i in solution is given by... 

The explicit (Euler) method described above has this stability limitation. There are other methods that do not. One of them (reverting again to the ode 21) is the backward difference (or backward implicit, BI) formula ... 

This equation can be solved either analytically [55, 58] or numerically [65-68] using the Backward Implicit approach [69-71] for the case of macroelectrodes at steady... 

The backwards implicit approximation, whose solution requires, in principle, an approach... 

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. 

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