Backward difference-methods

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. 

As the local error is the same as in the forward difference method, i.e. 0 k) + OQp-), the error from the time discretization dominates, assuming step sizes and h k. Although the method is unconditionally stable, its accuracy is low due to the large local truncation error. Overall, the backward difference method is 0 k), which means that the error at a certain point decreases linearly with k. In other words, if k is cut in half, so is the error. In contrast, if h is cut in half (for a given K), the error would be about the same, and the amount of computation would increase. 

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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 mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

The kinetic parameters in the rate equations were determined with non-linear regression analysis. The rate equations were inserted into the mass balances, which were solved numerically with the backward difference method during the parameter estimation. In the parameter estimation, the following objective function was minimized ... 

Since the time derivative used in the Crank-Nicolson method is second order correct, its step size can be larger and hence more efficient (see Fig. 12.11c). Moreover, like the backward difference method, the Crank-Nicolson is stable in both space and time. 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

The weighted sum of squares between measured and estimated concentrations was minimized by a hybrid simplex-Levenberg-Marquardt algorithm implemented in the simulation software Modest (Haario 1994). The model equations were solved in situ during the parameter estimation by the backward difference method. The estimated parameters were the kinetic and adsorption equilibrium constants of the system. The simulation results revealed that the model was able to describe the behaviour of the system. The parameter values were reasonable and comparable with values obtained in previous studies concerning citral hydrogenation in a slurry reactor (Tiainen 1998). 

The conventional Galerkin method is employed to discrete governing equations. The backward difference method is used to discrete the time dependent term. After the pressure fields determined, the velocity can be updated then the average stresses can be calculated. 

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. 

This equation must be solved for y The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL(Ref. 224). 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

Equation (5.38) can be interpreted as the scalar product of a forward-moving density and a backward-moving time-dependent operator. The optimal field at time t is determined by a time-dependent objective function propagated from the target time T backward to time t. A first-order perturbation approach to obtain a similar equation for optimal chemical control in Liouville space has been derived in a different method by Yan et al. [28]. 

The governing equations of the model are discretized in space by means of the finite element method [3, 18], and in time through a fully implicit finite difference scheme (backward difference) [18], resulting in the nonlinear equation set of the following form, [4, 7],... 

The solution methodology of the determinants is similar to that of the well-known Thomas algorithm used for the numerical solution of a differential equation with the finite-difference method [50]. An essential difference from the Thomas algorithm is that the first step ofthe algorithm here is a so-called backward process. This means that the calculation of T starts from the last sublayer, that is, from the Mth sublayer ofthe determinant and it is continued down to the 1st sublayer. Thus, the value of Ti is obtained directly, in the fist calculation step. Then, applying the known value of Ti, the value of Pi can be obtained by means of the fist boundary condition at X= 0, namely ... 

The species balance relation Eq. 13.2-8 is transformed to a difference equation using the forward difference on the time derivative and the backward difference on the space derivative. The finite difference form of the x-momentum equation (Eq. 13.2-25) is obtained by using the forward difference on all derivatives, and is solved by the Crank-Nicolson method. The same is true for the energy equation (Eq. 13.2-26). 

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 implicit backward-difference algorithm does not show the stability problems encountered in the case of the explicit forward-difference method, and this results immediately by analyzing the expression of the amplification factor ... 

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... 

When applied to the solution of odes, the BI method (Chap. 4) uses a backward difference for the derivative on the left-hand side of (8.9) and the argument of the function on the right-hand side is the future, unknown, concentration vector. In our notation, at the point along the row of concentrations, this is... 

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