Forward difference first derivative

For the second derivative, we note that this means derivative of the first derivative or rate of change of slope, at. We have two slopes about P - the forward and backward difference first derivatives - and it... 

This approximation is called a forward difference since it involves the forward point, z + Az, as well as the central point, z. (See Appendix 8.2 for a discussion of finite difference approximations.) Equation (8.16) is the simplest finite difference approximation for a first derivative. 

Besides, the potential and its first derivatives are continuous at boundaries where a volume density is discontinuous function. It is obvious that in this case the solution of the forward problem is unique. Now consider a completely different situation, when a density 5 q) is given only inside some volume V surrounded by a surface S, Fig. 1.8a. Inasmuch as the distribution of masses outside V is unknown, it is natural to expect that Poisson s equation does not uniquely define the potential U, and in order to illustrate this fact let us represent its solution as a sum ... 

When the user, whether working on stand-alone software or through a spreadsheet, supplies only the values of the problem functions at a proposed point, the NLP code computes the first partial derivatives by finite differences. Each function is evaluated at a base point and then at a perturbed point. The difference between the function values is then divided by the perturbation distance to obtain an approximation of the first derivative at the base point. If the perturbation is in the positive direction from the base point, we call the resulting approximation a forward difference approximation. For highly nonlinear functions, accuracy in the values of derivatives may be improved by using central differences here, the base point is perturbed both forward and backward, and the derivative approximation is formed from the difference of the function values at those points. The price for this increased accuracy is that central differences require twice as many function evaluations of forward differences. If the functions are inexpensive to evaluate, the additional effort may be modest, but for large problems with complex functions, the use of central differences may dramatically increase solution times. Most NLP codes possess options that enable the user to specify the use of central differences. Some codes attempt to assess derivative accuracy as the solution progresses and switch to central differences automatically if the switch seems warranted. 

Using 4 grid points to represent d/dx Instead of using the first order (backward or forward) or the second order (central) finite difference approximation for the first derivative, let us calculate the derivative using four grid points (see Fig. 8.2)... 

The value of the first derivative depends on the position at which it is evaluated. Setting x = +Ax gives a second-order, forward difference. ... 

It is apparent that the central difference approximations converge 0(Ax2). The forward and backward approximations to the first derivative converge O(Ax). This is because they are really approximating the derivatives at the points x = Ax rather than at x = 0. 

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... 

Analogous to the forward-difference method previously discussed, it is only first-order accurate in At. The only formal difference with respect to the forward-difference equation (8-10) appears to be the fact that the space derivative is evaluated at time tn+i, not at time tn. 

In this explicit scheme, the first-order forward difference approximation is used for the time derivative. The second-order central difference approximation is used for the spatial derivatives. Hence, the finite difference equation (FDE) of the partial differential equation (PDE) Eq. (10.2) is... 

Using Taylor series expansion, find the forward second-order accurate finite difference expansion for the first derivative of the... 

We now have three two-point approximations for a first derivative, all in fact being the same expression, (y2 — Vi)/h, but depending on where this formula is intended to apply, being, respectively a forward difference of 0(h) if applied at xt, a backward difference of 0(h) if applied at x2 and a central difference of 0(h2) if applied at (.iq +. r2)/2. In subsequent chapters, all these will be used to approximate, among others (2.3)-(2.8). 

In this way, the coefficients for any y((n) can be calculated. Table A.l in Appendix A shows them all, as whole numbers m/3j, where m is the multiplier mentioned above. For each n, the Table shows forward differences (at index 1), backward derivatives (at index n) and derivatives applying at points between the two ends. For n up to 6, all possible forms are included, as they will be needed later, while for n = 7, only the forward and backward formulae are shown, as only these are needed. In case the reader wonders why all this is of interest the forms y[(n) will be used to approximate the current in simulations (see the next section) the backward forms y n(n) will be used in the section on the BDF method in Chaps. 4 and 9, and the intermediate forms shown in the Table will be used for the Kimble White (high-order) start of the BDF method, also described in these chapters. The coefficients have a long history. Collatz [169] derived some of them in 1935 and presents more of them in [170]. Bickley tabulated a number of them in 1941 [88], The three-point current approximation, essentially y((3) in the present notation, was first used in electrochemistry by Randles [460] (preempted by two years by Eyres et al. [225] for heat flow simulations), then by Heinze et al. [301], and schemes of up to seven-point were provided in [133]. 

Two considerations regarding truncation error that enter into the derivation of the partial difference equations should be pointed out. In some published formulations of these equations, the first radial derivative has been approximated by a forward-difference expression (Kl, S5, Wl). This unsymmetrical formula has no advantage over a symmetrical or central-difference expression, but has a greater—lower order—truncation error. The central-difference approximation... 

The writing of O (Ax2) indicates that the discretisation error is proportional to Ax2 and therefore by reducing the mesh size the error approaches zero with the square of the mesh size. The first derivative with respect to time is replaced by the relatively inaccurate forward difference quotient... 

The first order derivative is given by the following finite difference approximations from Fq. (1) forward difference... 

Solution of these two equations yields second-order-correct forward difference expressions for first and second derivatives of /(x) at x = xq. The desired result for the first derivative is... 

As mentioned, the main challenge in this problem is the development of a stable, consistent procedure to handle the coupled, nonlinear boundary conditions at the interface ((18.12) and (18.13)). More specifically, if we regard (18.13) as an expression for Pg, then an approximation for the derivative D(f> /Dt is required. We have found that for a wide array of problems, it is adequate to approximate this derivative using a first-order forward difference scheme... 

From these equations corresponding formulas for forward difference and backward difference expressions of the first derivative can be easily drawn out ... 

We can now compare the approximated expressions for f and f" derived from Taylor series expansions with the finite difference formulation of the II Fick s law derived by following a physical approach. The first derivative with respect to time, da/duis approximated by a forward difference formula ... 

In molecular dynamics (MD) simulation atoms are moved in space along their lines of force (which are determined from the first derivative of the potential energy function) using finite difference methods [27, 28]. At each time step the evolution of the energy and forces allow the accelerations on each atom to be determined, in turn allowing the atom changes in velocities and positions to be evaluated and hence allows the system clock to move forward, typically in time steps of the order of a few fs. Bulk system properties such as temperature and pressure are easily determined from the atom positions and velocities. As a result simulations can be readily performed at constant temperature and volume (NVT ensemble) or constant temperature and pressure (NpT ensemble). The constant temperature and pressure constraints can be imposed using thermostats and barostat [29-31] in which additional variables are coupled to the system which act to modify the equations of motion. 

Alternative representations for the first derivative are the first-order accurate backward and forward difference formulas... 

Discretization of model equations refers to the approximation of (usually) the first and higher order derivatives in the model. There are several difference approximations that can be made, a forward difference approximation, a backward and a central difference approximation. A backward approximation can be written as ... 

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