Integration Gauss formulae

Now let us transform the surface integral in (13.133) into a volume integral, using the Gauss formula. According to equation (13.15), the normal stresses, P , can be expressed by the stress tensor as follows ... 

Substituting expression (13.134) into the surface integral and applying the Gauss formula, we find ... 

Many algorithms have been proposed to perform the numerical integration of functions. We consider only two femilies of algorithms, which are the basis for the development and implementation of an even number of general programs for the numerical integration the Newton-Cotes and the Gauss formulae. 

These algorithms are frequently stated in terms of integrals over [-1,1], termed Gauss-Legendre quadrature, and the general formula then is... 

This reduces (154) to the Christoffel formula for the Gauss numerical integration/quadrature [45] ... 

Although it is, in principle, possible to evaluate (6.2.11) analytically, it is numerically more advantageous to evaluate the integral in (6.2.11) with the help of a suitable Gauss-Laguerre integration formula (see, e.g., Stroud and Secrest (1966)). 

As for the pure diffusion problem, the key step of the finite volume method is the integration of the governing equations over the grid cell volume to yield a set of discretized equations at the nodal point P. By use of the Gauss theorem and the midpoint quadrature formula, we obtain ... 

Substituting (5.177)-(5.179) into (5.173) and transforming the surface integral into the volume integral according to the Gauss-Ostrogradsky formula (5.1), one obtains... 

Since the Gauss-Lobatto rules are based on a close formula, functions that cannot be evaluated at the interval extremes cannot be integrated. 

Some recent Runge-Kutta formulae are based on quadrature methods, that is, the points at which the intermediate stage approximations are taken are the same points used in integration with either Gauss or Lobatto or Radau rules (Chapter 1). For example, the Runge-Kutta method derived from the Lobatto quadrature with three points (also called the Cavalieri-Simpson rule) is... 

The Rys-Gauss quadrature developed by Dupuis, Rys, and King in the late 1970s is a method which at first glance seems to have little in common with the other integral schemes. " However, as can be demonstrated the formulae derived for the Rys-Gauss quadrature are connected to the formulae of the incomplete Gamma function based schemes. 

The computation of the incomplete Gamma functions is a vital part of evaluating of the ERIs for all integral methods except the Rys-Gauss quadrature. In the evaluation of Fm(T ) two formulae can be used depending on the value of the argument, T. Firstly, there is the asymptotic formula in which the Laplace formula (see equation 13) is utilized. 

A more accurate and systematic way of evaluating integrals is to perform the integration numerically. In the next two sections, we derive Newton-Cotes integration formulas for equally spaced intervals and Gauss quadrature for unequally spaced points. 

Example 4.4 Integration Formulas - Gauss-Legendre Quadrature. Write a general... 

Numerical differentiation and integration, including differentiation by backward, forward, and central finite differences Newton-Cotes formulas and the Gauss Quadrature... 

Error analysis is not so simple. The above formula is exact if fix) is a cubic polynomial (or a simpler one). A rule of thumb is that the order of accuracy of Gauss integration is twice that of equally spaced methods using the same number of data points. 

This example illustrates the power of Gauss integration. A four-point formula gives essentially the exact result. [This is due to the fact that f x) is a quartic function, and the four-point formula is exact for polynomials up to degree 7.]... 

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