Rosenbrock Methods

The resulting system is called a set of differential-algebraic equations (DAE) and their solution is now a specialised field with its own texts [130, 286] and there is a package program, DASSL [441], for their solution. This can be of use in the present context, for example with the method of lines, which indeed often results in a DAE system. This is gone into in some detail in Chap. 9, in the context of Rosenbrock methods. 

This section is left to the last because it pertains to systems of odes (or DAEs), although Rosenbrock methods are a kind of Runge-Kutta method... 

Before moving on to real Rosenbrock methods, consider again (4.66). The left-hand side contains a term in fy if we are dealing with a system of odes, this is called the Jacobian of the system. It is often constant, evaluable in advance. It will be seen in Chap. 9 that unless the diffusion problem has nonlinear concentration terms (for example from higher-order homogeneous reactions), the Jacobian is constant. If not, it must be evaluated at every step. 

In general, a Rosenbrock method consists of a number s of stages. At each stage, a Runge-Kutta-type ki value is calculated, from explicit rearrangement of implicit equations for these. At stage i, the equation is... 

There are, however, implicit variants of RK, and these may have promise. There are several classes of these, see a thorough text on the subject [284,286]. One of these classes, the Rosenbrock method, has been recently examined [100,113, and see references therein] and found very efficient. This is described in its own Sect. 9.4, below. 

For the basics of this method, see Chap. 4. There it was mentioned that Bieniasz introduced this method to electrochemical simulation [100], preferring ROWDA3, a third-order variant that also has a smooth response. There exists a second-order variant with a smooth response, ROS2, due to Lang [347], which might be more appropriate if second-order spatial derivative approximations are to be used. Coefficients for some variants are given in Appendix A. The object here is to describe the way Rosenbrock methods are used in the present context. The Bieniasz paper [100] shows the way (but the standard symbols, as used in Chap. 4, are used here, rather than those used by Bieniasz). 

We are now ready to invoke the Rosenbrock method. A number s of fcj vectors must be computed, s being the order chosen. The general equation for each one is an extension of that given for a pure ode set on page 70, (4.70), to the present DAE case, introducing the selection matrix S and following Bieniasz [100] (though with the more common notation) ... 

An obvious alternative choice of method, given the probably nonlinear form of the isotherm boundary condition is to use a Rosenbrock method. Then, the two boundary conditions are simply the first two equations in a whole DAE set, the first of the pair (10.3) being an algebraic equation, the second an ode. The Rosenbrock method is described in Chap. 9, Sect. 9.4 starting on page 167. 

As for the transport- and isotherm-controlled case above, these equation sets can now be handled either using a standard implicit method or, perhaps logically in the case of a nonlinear isotherm, a Rosenbrock method. 

Simulations must thus handle the nonlinear boundary conditions. Some have taken the easy way out and used explicit methods [123,429]. Bieniasz [105] used the Rosenbrock method (see Chap. 9), which makes sense because it effectively deals with nonlinearities without iterations at a given time step. 

OC Chap. 9, Sect. 9.6. Produces impressively accurate results with only a few spatial points. It can be regarded as a kind of MOL, and although practitioners tend to propose advanced techniques for solving the set of odes, other methods can be used, such as a Rosenbrock method or BDF. The method is not used as much as one might expect. 

The Rosenbrock method is described for odes in Chap. 4 and for electrochemical simulations, that is, DAEs, in Chap. 9. There are four variants, two of... 

Rosenbrock methods require a set of constants, and as these are needed in several subroutines in a given program, it is convenient to gather them in a... 

Identification of the theoretical and experimental transfer functions in order to estimate the effective diffusivity De is obtained by minimizing a relative error function taken between the two transfer functions. The Rosenbrock method of optimization has been used. All the measurements have been made at room temperature and something close to normal atmospheric pressure. The only parameter that changes is the carrier gas flow rate. 

Below, we describe four algorithms that are able to handle small and medium dimension problems even in the presence of relatively narrow valleys without using any gradient or Hessian the Rosenbrock method (1960), the Hooke-Jeeves method (1961), the Simplex method (Spendley et al, 1962 Nelder and Mead, 1965), and the Optnov method (Buzzi-Ferraris, 1967). Note that their current structure is slightly different from the original one. 

The Rosenbrock method is slightly modified using this technique the first two points are still the same, but it is necessary to continue with the following ones ... 

The Rosenbrock method (and its variants) has the following pros and cons. 

It is important to realize why even the Rosenbrock method becomes inefficient when the function valleys are particularly narrow. 

Figure 3.3 The Rosenbrock method fails with very narrow valleys.

Roche, M., Rosenbrock methods for differential algebraic equations. Nu mer. Math. 52, 45-63 (1988)... 

For our chemical problem, we successfully use the stiff versions of the solvers LSODE (from A.C.Hindmarsh) and VODE (from G.D.Byrne and A.C.Hindmarsh), which employ multistep methods (backward differentiation formulas) and allow to change stepsize and order of the methods. Comparing investigations show that the IVP-solver RODAS (from E.Hairer and G.Wanner), an implementation of onestep methods (Rosenbrock methods), also with variable stepsize, is working with same success (see e.g. [4, 5]). 

In contrast with algorithms using univariate search, the Rosenbrock method is a so-called acceleration method, which makes the direction and/or the distance (in this case both) of the parameter jumps dependent on the degree of success of the previous parameter jumps. With p parameters, the algorithm proceeds as follows ... 

Fig. 9.2 illustrates the Rosenbrock method for a model with two parameters. Because the algorithm attempts searching along the axis of a valley of the objective function S, many evaluations of S can be omitted compared with, for example, the algorithm with univariate... 

The Rosenbrock method applied to the objective function of a regression analysis with a model consisting of two parameters, and 2- The figure shows the first stage of the optimization. 

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