Gauss-Radau method

A higher-order, nonsymplectic method used to integrate Newton s equations of motion is the Gauss-Radau method. " Similarly to the Runge-Kutta method, this technique divides the time step. At, into substeps, h, and, by using the forces that are evaluated at each h, provides accurate integration of the positions and momenta. The mass weighted force (i.e., acceleration) over a time step is expanded in the substeps, h, as... 

Another important family of methods is the set of Gauss-Radau formulae. In this family, just one of the two extremes of interval is a support point They are... 

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). 

A second feature of this method is that the substeps, h, are not equally spaced over At. Implementation of Gauss-Radau spacings allows for the cancellation of higher-order terms in equation (23). For example, seventh-order accuracy (which would require the inclusion of terms to Ar if h were equally spaced) can be achieved with explicit consideration of terms up to fourth order only. This facilitates accurate and efficient integration. 

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... 

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