Lobatto method

For other examples of Gauss-Lobatto methods of different orders, we refer readers to web databases. In Section 1.10.3, we will demonstrate how to implement these algorithms to obtain a very efficient algorithm. 

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). 

The purpose of this paper is to construct an improved automatic quadrature by the proper combination of Ninomiya s method and the FLR one. By using Kahaner s twenty one problems [9] we compare in performance the present method with Ninomiya s method, the FLR method and an adaptive Lobatto procedure with Kronrod extension due to Gander and Gautschi [8]. We also give the comparison results with Venter-Laurie s method [16] for 32 test problems. 

GLL Gauss-Lobatto-Legendre collocation points CM RES Generalized Minimal RESidual method IGCC Integrated Gasification Combined Cycle... 

Algorithms Derived from the Trapezoid Method 9 Table 13 Values of x,- and w, for the Gauss-Lobatto formula with five points. 

A third femily of methods of relevant interest is the Gauss-Lobatto. In this family, the two extremes of the interval are support points and, thus, the formulae are close. 

Method Derived from the Gauss-Lobatto Algorithm... 

The adaptive methods that use the Gauss-Lobatto algorithms have the following pros and cons. 

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... 

Lobatto s method with three points is similar to Cavalieri-Simpson s method, where the three points are at the extremes and in the center of the interval. The corresponding Runge-Kutta s method is... 

We shall show here that if just two temperatures, chosen at "Lobatto points", are used to characterize the transverse temperature distribution in a laminar lubricating film, the effects of that distribution can be surprisingly well predicted. The calculations we have 80-far performed have been directed solely towards demonstrating this fact, and only rudimentary numerical methods have been used in the plane of the film. Accordingly, no meaningful computation times can yet be reported. But we believe the technique will prove to be quite suitable for practical calculations. 

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