Numerical methods Taylor series expansion

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

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. ... 

Euler s methods can be derived from a more general Taylor s algorithm approach to numerical integration. Assuming a first-order differential equation with an initial value such as [dy/dx] = / = function of x, and y = f(x,y) with y(xo) = yo. if the f(x,y) can be differentiated with respect to x and y, then the value of y at X = (xo + h) can be found from the Taylor series expansion about the point x = xq with the help ofEq. (16) ... 

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. 

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 calculation of force constants at points along the MEP is often done separately from the determination of the path by numerical integration of Eq (2), but these two problems can profitably be combined. Methods recently have been proposed [6,7] that efficiently use the available force constants to better follow the path. To understand these methods and the relationship between them, consider two different Taylor series expansions about a point on the MEP. The first is the familiar expansion of the energy in the mass-weighted Cartesian coordinates,... 

Eq. 6.2.6 was solved analytically to obtain the operation curve of the reactor (X vs t). Lumped kinetic parameters were determined by non-linear regression of experimental data using the numerical method of Newton-Raphson with first-order Taylor series expansion. Lumped parameters were smooth functions of temperature all parameters were adequately fitted to second order polynomials except for D that required a fourth order polynomial. The model can be used for reactor temperature optimization and can be extended to prolonged sequential batch operation provided that a sound model for enzyme inactivation is validated (Illanes et al. 2005b). 

The use of quantum-chemistry computer codes for the determination of the equilibrium geometries of molecules is now almost routine owing to the availability of analytical gradients at SCF, MC-SCF and CP levels of theory and to the robust methods available from the held of numerical analysis for the unconstrained optimization of multi-variable functions (see, for example. Ref. 21). In general, one assumes a quadratic Taylor series expansion of the energy about the current position... 

Finite-difference schemes involve some form of truncated Taylor series expansion. The finite-element technique uses a local basis function to minimize numerical error, while the spectral method utilizes global basis functions. A spectral method has the advantage that differential relations are converted to algebraic expressions. The semi-Lagrangain scheme is based on fitting interpolation... 

In the method using Taylor series expansion as explained in previous sections, the accuracy of the robustness evaluation depends directly on the reliability of the numerical sensitivity analysis. For this reason, when the evaluation of numerical sensitivities has some difficulties resulting from the elastic-plastic structural property of isolators, another URP method should be introduced where the variation of the objective function is... 

The first-order second moment method (FOSM) is the method adopted within the framework to propagate input parameter uncertainty through numerical models (26, 27). FOSM provides two moments, mean and variance of predicted variables. This method is based on Taylor series expansion, of which second-order and higher terms are truncated. The expected value of concentration, E[u] and its covariance, COV[u] are (25, 27),... 

We first consider Newton s method, an iterative technique that is based on the use of Taylor series expansions. As Taylor series are used extensively in numerical mathematics, we briefly review their use. 

In the two previous chapters the code segments developed, in particular new-ton() and nsolv() used a numerical derivative for the first-order Taylor series expansions of various functions. In these routines the simple single sided derivative equation was used with a relative default displacement factor of 10 . The single sided derivative was used because in all the routines involving Newton s method, the funetion value is required in addition to the derivative and thus the single sided derivative requires only one additional funetion evaluation whereas the use of the... 

To start, we substitute the series expansion (Eq. 6.5) into the governing equation and the boundary condition (if the equation is a differential equation), and then apply a Taylor expansion to the equation and the boundary condition. Now, since the coefficients of each power of e are independent of e, a set of identities will be produced. This leads to a simpler set of equations, which may have analytical solutions. The solution to this set of simple subproblems is done sequentially, that is, the zero order solution y ix) is obtained first, then the next solution y,(x), and so on. If analytical solutions cannot be obtained easily for the first few leading coefficients (usually two), there is no value in using the perturbation methods. In such circumstances, a numerical solution may be sought. 

A method similar to the iterative, is the partial closure method [37], It was formulated originally as an approximated extrapolation of the iterative method at infinite number of iterations. A subsequent more general formulation has shown that it is equivalent to use a truncated Taylor expansion with respect to the nondiagonal part of T instead of T-1 in the inversion method. An interpolation of two sets of charges obtained at two consecutive levels of truncations (e.g. to the third and fourth order) accelerates the convergence rate of the power series [38], This method is no longer in use, because it has shown serious numerical problems with CPCM and IEFPCM. 

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