Interpolation and finite differences

In the Verlet method, this equation is written by using central finite differences (see Interpolation and Finite Differences ). Note that the accelerations do not depend upon the velocities. 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

Other analysis methods that use discretization include the finite difference method, the boundary element method, and the finite volume method. However, FEA is by far the most commonly used method in structural mechanics. The finite difference method approximates differential equations using difference equations. The method works well for two-dimensional problems but becomes cumbersome for regions with irregular boundaries (Segerlind 1984). Another difference between the finite element and finite difference methods is that in the finite difference method, the field variable is only computed at specific points while in the finite element method, the variation of a field variable within a finite element is available from the assumed interpolation function (Hutton 2003). Thus, the derivatives of a field variable can be directly determined in finite element method as opposed to the finite difference method where only data concerning the field variable is available. The boundary element method is also not general in terms of structural shapes (MacNeal 1994). 

If the collocation points are chosen at equidistant intervals within the interval of integration, then the collocation method is equivalent to polynomial interpolation of equally spaced points and to the finite difference method. This is not at all surprising, as the development of interpolating polynomials and finite differences were all based on expanding the function in Taylor series (see Chap. 3). It is not necessary, however, to choose the collocation points at equidistant intervals. In fact, it is more advantageous to locate the collocation points at the roots of appropriate orthogonal polynomials, as the following discussion shows. 

Comparison of Alignment Charts and Cartesian Graphs. There are typically fewer lines on an alignment chart as compared to Cartesian plots. This reduces error introduced by interpolation and inconsistency between scales. For example, to find a point (x,j) on a Cartesian graph one draws two lines, one perpendicular to each axis, and these reference lines intersect at the point x,j). This point (x,j) may correspond to some finite value found by rea ding a contour map represented by a family of curves corresponding to different values of the function. 

The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-80) to provide the Galerkin finite element equations. For example, with the grid shown in Fig. 3-48, a linear interpolation would be used between points x, and, vI+1. 

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

Two different formulas for quadratic interpolation can be compared Equation (5.8), the finite difference method, and Equation (5.12). 

Determine the relative rates of convergence for (1) Newton s method, (2) a finite difference Newton method, (3) quasi-Newton method, (4) quadratic interpolation, and (5) cubic interpolation, in minimizing the following functions ... 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

These equations are now in convenient form for a finite-difference scheme along lines similar to that used above. Ail alternative approach developed in some detail employs the Lagrangian interpolation formula to follow the motion of the boundary in the (x/a, r) plane. This is a means of developing finite-difference approximations to derivatives based on functional values and not necessarily equally spaced in the argument. Crank points out that the application of Lagrangian interpolation formulas involves a relatively large number of steps in time, whereas the fixedboundary procedures require iterative solutions at each time interval, which are, however, far fewer in number. 

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

Here, we obtained a backward finite difference expression for the derivative d/dx at a point i, and we get the order of the interpolation, i.e., all the terms that we collapsed in 0(Ax), which is a metric for the error built into the approximation. Similarly, we can obtain the forward difference approximation as... 

An improved 0(h2) finite-difference representation of the boundary condition (8-44) results by approximating the solution in the vicinity of the boundary by the second order Lagrange interpolating polynomial passing through the points (xi,cj), (x2,cg), and (x3,cg) (equally spaced gridpoints are assumed) ... 

The numerical Method of Lines as implemented in the routine NDSolve of the Mathematica system deals with system (32) by employing the default fourth order finite difference discretization in the spatial variable Z, and creating a much larger coupled system of ordinnary equations for the transformed dimensionless temperature evaluated on the knots of the created mesh. This resulting system is internally solved (still inside NDSolve routine) with Gear s method for stiff ODE systems. Once numerical results have been obtained and automatically interpolated by NDSolve, one can apply the inverse expression (31.b) to obtain the full dimensionless temperature field. 

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