Difference backward

Equally Spaced Backward Differences Backward differences are defined by... 

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). 

Approximate expressions for derivatives of any order are given in terms of forward and backward difference expressions as... 

Coefficients of forward difference expressions for derivatives of up to the fourth order are given in Figure 1-52 and of backward difference expressions in Figure 1-53. 

A central difference expression may be derived by combining the equations for forward and backward differences. 

Given a data table with evenly spaced values of x, and rescaling x so that h = one unit, forward differences are usually used to find f(x) at x near the top of the table and backward differences at x near the bottom. Interpolation near the center of the set is best accomplished with central differences. 

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... 

Setting A = A gives a second-order, backward difference. ... 

Apply finite difference approximations to Equation (9.15) using a backwards difference for da/d and a central difference for d a/d. The result is... 

The marching equation for reverse shooting. Equation (9.24), was developed using a first-order, backward difference approximation for dajdz, even though a second-order approximation was necessary for (faldz. Since the locations j—l, j, and j+ are involved an5rway, would it not be better to use a second-order, central difference approximation for dajdz ... 

The best packages for stiff equations (see below) use Gear s backward difference formulas. The formulas of various orders are [Gear, G. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. (1971)]... 

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)]. 

As shown in this chapter for the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaced by their finite-differenced equivalents. These finite-differ-enced forms of the model equations are shown to evolve as a natural consequence of the balance equations, according to Franks (1967), and as derived for the various examples in this book. The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end-sections can be better approximated by the forward and backward differences as derived in the previous examples. The various forms of approximation based on the use of central, forward and backward differences have been listed by Chu (1969). 

Fourier channel k. The phase is known up to a multiple of 27t(since only exp /( ),(/")) is known). Time-scale modifications also require the knowledge of the instantaneous frequency G) (f ). 0), Uua) can also be estimated from successive short-time Fourier transforms for a given value of k, computing the backward difference of the short-time Fourier transform phase yields... 

Taylor-series expansions allow the development of finite differences on a more formal basis. In addition, they provide tools to analyze the order of the approximation and the error of the final solution. In order to introduce the methodology, let s use a simple example by trying to obtain a finite difference expression for dp/dx at a discrete point i, similar to those in eqns. (8.1) to (8.3). Initially, we are going to find an expression for this derivative using the values of

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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 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... 

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