Radau method

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). 

In Fig. 4.8 the stability region of the three stage Radau Ila method is displayed. One realizes that the stability of the Radau method is much alike the stability of the implicit Euler method, though the three stage Radau method has order 5. Again we note the property i (0) = 1 which corresponds to zero stability in the multistep case. 

Often, the term stiff differential equation is used to indicate that special methods are used for numerically solving them. These methods are called stiff integrators and are characterized by A-stability or at least i4(a)-stability. They are always implicit and require a corrector iteration based on Newton s method. For example BDF methods or some implicit Runge-Kutta methods, like the Radau method are stiff integrators in that sense. 

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... 

Figure. 1 Potential energy profiles together with some of the reactant energy levels for the H + H2O —> OH + H2 (a) and the H + H F —> F + H2 reaction (b). On panel (a) the energies of the stationary points are derived from the WSLFH PES [11], the initial energy levels for water are calculated with a DVR method using Radau coordinates, and the final levels for the products are obtained from the Morse potentials corresponding to the PES. On panel (b) the energies of the stationary points are obtained from the 6-SEC PES [13] both the initial and final energy levels are calculated from the Morse parameters obtained by fitting a Morse curve to the potential for the separated FIF and H2 oscillators.

To illustrate this method we consider in some detail the resonances of the 3D FHH system on the Muckerman l surface ( O) As noted in the previous section, collinearly the resonance may be identified with an RPO having Ah action. Defining the bend angle Y via the Natanson-Smith-Radau (62,63) coordinate system, we find the Ah action RPO at fixed y. This provides the curve E.(y) (shown in Fig. 7) where Ej (Y ) is the energy of the y dependent" RPO. 

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... 

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... 

For the above new family the authors describe the production methodology, they study the order conditions for this class of methods and also the stability properties. Numerical experiments give the advantage of the new class of methods (the method has lower computational cost in a fixed step size implementation and same behavior for the error as the indirect collocation Radau IIA method)... 

The polyspherical approach provides, for a particular class of curvilinear coordinates, general expressions for the elements of the G, C and F matrices, whatever the number of atoms and the set of - 1 vectors (Jacobi, Radau, valence,...) used to parametrize the system. In addition, the use of a specific definition for the BF frame ensures that the KEO has the so-called product form (see Sect. 4.2.2), i.e. it is expressed as a sum of products of operators acting on a single coordinate. This definition of the BF frame is as follows the axis is parallel to one of the — 1 vectors used to parametrize the system and the (xz) BF half-plane, with x > 0 is parallel to another vector. A detailed presentation of the method can be found in Ref. [6]. 

The coefficient matrix of the three stage Radau Ila method (4.3.5) has one real... 

By this theorem the continuous representation of the fifth order Radau Ila method has order three. 

As for stiffly accurate methods, like Radau IIA methods,... 

Example 5.4.2 We discretize the linearized constrained truck with a 3-stage Gaufi method and a 3-stage Radau IIA method. The eigenvalues of the discrete-time transfer matrix with... 

Table 5.5 Eigenvalues of the discrete linear truck example in its index-2 formulation for the three stage GauB and Radau Ila methods, h = 0.1.

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. 

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