Taylor series finite differences

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Interestingly, if one Taylor series expands Eq. (36) and equates the terms of the same order in kj with Eq. (37) one can derive the standard Lagrangian FD approximations (i.e., require the coefficient of kj to be —1, and require the coefficient of all other orders in kj up to the desired order of approximation to be 0.) A more global approach is to attempt to fit Eq. (36) to Eq. (37) over some range of Kj = kjA values that leads to a maximum absolute error between Eq. (36) and Eq. (37) less than or equal to some prespecrfied value, E. This is the essential idea of the dispersion-fitted finite difference method [25]. 

By replacing the derivatives in the transport equation with finite difference approximations, we introduce to our numerical solution specific inaccuracies. To see this, we write a Taylor series expanding the difference between and... 

Comparing this equation to a finite difference approximation (Eqn. 20.29), we see that in the numerical solution we carry only the first term in the series, the d Q /3x term, omitting the higher order entries. The Taylor series is truncated, then, and the resulting error called truncation error. 

The derivatives of the Taylor series are all finite. It is not necessary to expand the series at xB = 0, but it is most common and convenient for dilute solutions. The Taylor series expansion of In yB may be expressed in a different notation as... 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

All numerical errors that arise from or are related to the discretization, i.e. representing conservation equations in discrete form using for example finite elements or finite differences, can be expressed by a Taylor series (Richarson, 1910)... 

Typically, these methods arrive at the same finite difference representation for a given problem. However, we feel that Taylor-series expansions are easy to illustrate and we will therefore use them here in the derivation of finite difference equations. We encourage the student of polymer processing to look up the other techniques in the literature, for instance, integral methods and polynomial fitting from Tannehill, Anderson and Pletcher [26] or from Milne [16] and finite volume approach from Patankar [18], Versteeg and Malalasekera [27] or from Roache [20]. 

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

backward difference equation). Thus, we are looking for an expression such as... 

Second order finite difference for a second order derivative. Let s illustrate the Taylor-series by finding a finite difference for the second derivative... 

The independent variable in ordinary differential equations is time t. The partial differential equations includes the local coordinate z (height coordinate of fluidized bed) and the diameter dp of the particle population. An idea for the solution of partial differential equations is the discretization of the continuous domain. This means discretization of the height coordinate z and the diameter coordinate dp. In addition, the frequently used finite difference methods are applied, where the derivatives are replaced by central difference quotient based on the Taylor series. The idea of the Taylor series is the value of a function f(z) at z + Az can be expressed in terms of the value at z. 

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

It was soon realised that at least unequal intervals, crowded closely around the UMDE edge, might help with accuracy, and Heinze was the first to use these in 1986 [300], as well as Bard and coworkers [71] in the same year. Taylor followed in 1990 [545]. Real Crank-Nicolson was used in 1996 [138], in a brute force manner, meaning that the linear system was simply solved by LU decomposition, ignoring the sparse nature of the system. More on this below. The ultimate unequal intervals technique is adaptive FEM, and this too has been tried, beginning with Nann [407] and Nann and Heinze [408,409], and followed more recently by a series of papers by Harriman et al. [287,288,289, 290,291,292,293], some of which studies concern microband electrodes and recessed UMDEs. One might think that FEM would make possible the use of very few sample points in the simulation space however, as an example, Harriman et al. [292] used up to about 2000 nodes in their work. This is similar to the number of points one needs to use with conformal mapping and multi-point approximations in finite difference methods, for similar accuracy. 

Expanding these two flows as functions of their conjugate forces in a Taylor series about some reference steady state, and assuming all other forces as constant, yields the finite differences from the first-order terms... 

This approximate expression of the derivative in terms of differences is the finite difference form of the first derivative. The equation above can also be obtained by writing the Taylor series expansion of the function/about the point. t. 

C How is the finite difference formulation for the first derivative related to the Taylor series expansion of the solution function ... 

The equations (2.238) and (2.239) for the replacement of the derivatives with difference quotients can be derived using a Taylor series expansion of the temperature field around the point (Xi,tk), cf. [2.53] and [2.57]. It is also possible to derive the finite difference formula(2.240) from... 

Assume that we have a continuous ftmction, G(x), and that we use a Taylor series expansion to calculate its first and second differentials, using a finite difference scheme involving the values of G(z) at Xj i, Xj, and Xi+. We let Gj = G(x,), and write the Taylor expansions for G,+i and G, i,... 

The finite difference approach is a widely used discretization technique because of its simplicity. Finite difference approximations of derivatives are obtained by using truncated Taylor series. The following Taylor expansions can be used ... 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

Any approximation of the derivative of a function in terms of values of that function at a discrete set of points is called a finite difference approximation. There are several ways of constructing such approximations, the Taylor series approach illustrated above is frequently used in numerical analysis because it supplies the added benefit that information about the error is obtained. Another method uses interpolation to provide estimates of derivatives. In particular, we use interpolation to fit a smooth curve through the data points and differentiate the resulting curve to get the desired result. A collection of low order approximations (i.e., first to fourth order polynomial approximations) of first and second order derivative terms can be found in textbooks like [49, 50, 167]. 

Here G is a time-dependent, experimental variable, and the space vector r(f) is time dependent because of translational motion of the nuclear spins (cf. Section 1.2). Therefore, the phase 0 is time-dependent as well. For short times, the final position r t) of the spins assumed after the time has elapsed, can be approximated by a Taylor series with a finite number of terms (cf. eqn (5.4.54)). These terms are discriminated by the power of the time lag t and involve initial position r(0), velocity v(0), and acceleration a(0) as coefficients to different moments mk of the time-dependent gradient vector G(t),... 

This section deals with the construction of optimal higher order FDTD schemes with adjustable dispersion error. Rather than implementing the ordinary approaches, based on Taylor series expansion, the modified finite-difference operators are designed via alternative procedures that enhance the wideband capabilities of the resulting numerical techniques. First, an algorithm founded on the separate optimization of spatial and temporal derivatives is developed. Additionally, a second method is derived that reliably reflects artificial lattice inaccuracies via the necessary algebraic expressions. Utilizing the same kind of differential operators as the typical fourth-order scheme, both approaches retain their reasonable computational complexity and memory requirements. Furthermore, analysis substantiates that important error compensation... 

The derivative (D) being approximated by the finite-difference operator (FD) to within a truncation error (TE) (or, discretization error). The foregoing mathematical consideration provides an estimate of the accuracy of the discretization of the difference operators. It shows that TE is of the order of (Ax)2 for the central difference, but only O(Ax) for the forward and backward difference operators of first order. Equations (4.41) and (4.42) involve 2 or 3 nodes around node i at x , leading to 2- and 3-point difference operators. Considering additional Taylor series expansions extending to nodes i + 2 and i - 2 etc., located at x + 2Ax and x. — 2Ax, etc., respectively, one may derive 4- and 5-point difference formulas with associated truncation errors. Results summarized in Table 4,8 show that a TE of O(Ax)4 can be achieved in this manner. The penalty for this increased accuracy is the increased complexity of the coefficient matrix of the resulting system of equations. 

In our perturbative treatment, we do not require an analytical form for G or V. Instead we expand these coordinate-dependent terms, just as we do the potential V, in a Taylor series expansion about the equilibrium configuration (45,46,49). In order to evaluate the expansion coefficients we use finite difference techniques. The advantage of this strategy is that it only requires the evaluation of G and V at specific molecular configurations. These contributions are evaluated by noting that G = BB and that V is a function of the G-matrix elements. Here B is the (3n — 6) X 3n B-matrix, whose elements are... 

Since the initial data point is / = 0 at W = 0, the next few data points very close to the inlet allow one to estimate (dx/dW)miet via finite-difference algebra. The number of terms that are retained in the Taylor series prior to truncation dictates the correctness of the finite-difference calculation and the number of data points required to approximate inlet via... 

Nonequispaced data pairs are available for the function f x) at three values of the independent variable jco, xi, and X2, where xi —xq = h and X2 — xi = j (i.e, h j). The objective of this summary of numerical analysis is to generate expressions for df/dx and d f/dx at x = xi. These results will be used in the numerical algorithms of Sections 23-4 and 23-5 to solve the mass transfer equation. The starting point to develop several formulas in numerical analysis is the Taylor series for /(x), expanded about one of the given data points (i.e., X = xi, for example), nth-order-correct finite-difference expression are obtained by including n + terms in the infinite series expansion for /(x). Hence, for second-order correct results (i.e., n = 2), three terms are necessary ... 

All of the information provided at the beginning of this section has been used. It should be obvious, now, that nth-order-correct finite difference expressions for various derivatives require Taylor series expansions that include n + terms with n unknowns, and n -1-1 known data pairs to determine n unknown derivatives. The function /(x) is expanded about one of the data pairs (i.e., evaluation of the zeroth-order leading term), and the remaining n data pairs are used for nontrivial... 

Here Ax is the distance along the x axis between adjacent nodal points. Such replacement of derivatives over coordinates x, y, z and t by equivalent finite differences results at computation in some error. The finite difference between C and C, maybe represented as Taylor series... 

Like all other meshless methods, the first step in GFD is to scatter nodal points in the computational domain and along the boimdary. To each node (point), a collection of neighboring nodes are associated which is called star. The number and the position of nodes in each star are decisive factors affecting the finite difference approximation. Particular node patterns can lead to ill-conditioned situations and ultimately degenerated solutions. Using the Taylor s series expansion, the value of any sufficiently differentiable smooth function u at the central node of star, uq, can be expressed in terms of the value of the same function at the rest of nodes, with i = 1,. .N where N is the total number of neighboring nodes in the star and is one less than the total number of nodes in it. In two dimensions, a second-order accurate Taylor series expansion can be written as... 

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