Radau quadrature

Therefore, to use the Radau quadrature with the exterior point ( = 1) included, the N interior collocation points are chosen as roots of the Jacobi polynomial with a = 1 and jS = 0. Once N -I- 1 interpolation points are chosen, the first and second order derivative matrices are known. 

The Gaussian quadrature presented in the previous section involves the quadrature points which are within the domain of integration (0,1). If one point at the boundary, either at jc = 0 or j = 1, is included in the quadrature formula, the resulting formula is called the Radau quadrature. 

Similarly, when the Radau quadrature formula is used with the boundary point at X = 0 included instead of x = 1, the Radau quadrature formula is... 

Heretofore, we have considered the quadrature formula for the integral of a function yix) with a weighting function W(x) = 1. The Radau quadrature formula for the following integral with the weighting function W(x) = x (l - x) ... 

Table E.S Quadrature Points and Weights for Radau Quadrature with the First End Point Included Weighting Function Wix) = 1...

Table E.6 Quadrature Points and We ts for Radau Quadrature with the Last End Point Included Weighting Function lF(jr) = jc (l — x) ...

The last section illustrated the Radau quadrature formula when one of the quadrature points is on the boundary either at a = 0 or a = 1. In this section, we present the Lobatto quadrature formula, which includes both boundary points in addition to the N interior quadrature points. The general formula for the Lobatto quadrature is... 

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... 

Of the most common implicit algorithms, the most useful ones are those adopting quadrature points, which are points used by the open Gauss method, semiopen Radau method, and the close Lobatto method (see Chapter 1). 

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