Implicit ODEs

Some reactor models require a more general structure than the ODE, dxjdt = fix, t). The nonconstant density, nonconstant volume semibatch and CSTR reactors in Chapter 4 are more conveniently expressed as differential-algebraic equations (DAEs). To address these models, consider the more general form of implicit ODEs... 

Deuflhard, P., Nowak, U., Extrapolation integrators for quasilinear implicit ODEs In P. Deuflhard, B. Engquist (eds.) Large Scale Scientific Computing. Progress in Scientific Computing 7, 37-50, Birkhauser (1987)... 

To explain the idea, we consider system (1.3.11) with T p) = I. As we discuss implicit ODEs later (see Ch. 5), we will assume for the moment a system without constraints and with the mass matrix being the identity or inverted to the right hand side of the differential equation. For details also concerning implementation of the approach we refer to [Eic91]. 

The equations of motion of unconstrained multibody systems are normally given as implicit ODE... 

In that particular case the system consists of differential equations and some nonlinear equations, often called in this context (somehow sloppy) algebraic equations. Consequently, this type of singularly implicit ODEs is also called a system of differential algebraic equations (DAEs). 

This indicates that after an initial overhead of 0.319 model runs to set up the algorithm, an additional 0.07 of a model-run was required for the computation of the sensitivity coefficients for each additional parameter. This is about 14 times less compared to the one additional model-run required by the standard implementation of the Gauss-Newton method. Obviously these numbers serve only as a guideline however, the computational savings realized through the efficient integration of the sensitivity ODEs are expected to be very significant whenever an implicit or semi-implicit reservoir simulator is involved. 

ODE solver. Relative to non-stiff ODE solvers, stiff ODE solvers typically use implicit methods, which require the numerical inversion of an Ns x Ns Jacobian matrix, and thus are considerably more expensive. In a transported PDF simulation lasting T time units, the composition variables must be updated /Vsm, = T/At 106 times for each notional particle. Since the number of notional particles will be of the order of A p 106, the total number of times that (6.245) must be solved during a transported PDF simulation can be as high as A p x A sim 1012. Thus, the computational cost associated with treating the chemical source term becomes the critical issue when dealing with detailed chemistry. 

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

Thus the implicit methods become slower and slower as the number of ODEs increases, despite the fact that large step sizes can be taken. Therefore plain old explicit Euler turns out to run faster than the impheit methods on many reahstically large problems, unless the stiffness of the system is very, very severe. We will talk more about this in Chap. 5. 

For these time periods, the ODEs and active algebraic constraints influence the state and control variables. For these active sets, we therefore need to be able to analyze and implicitly solve the DAE system. To represent the control profiles at the same level of approximation as for the state profiles, approximation and stability properties for DAE (rather than ODE) solvers must be considered. Moreover, the variational conditions for problem (16), with different active constraint sets over time, lead to a multizone set of DAE systems. Consequently, the analogous Kuhn-Tucker conditions from (27) must have stability and approximation properties capable of handling ail of these DAE systems. 

Equation (2.1) is an ODE system, and, since the values of the variables xi and X2 at t = 0 are provided, it is an initial value problem. By employing a small perturbation parameter 0 < e [Pg.12]

The common underlying principle in the approaches for characterizing the solvability of a DAE system is to obtain, either explicitly, or implicitly, a local representation of an equivalent ODE system, for which available results on existence and uniqueness of solutions are applicable. The derivation of the underlying ODE system involves the repeated differentiation of the algebraic constraints of the DAE, and it is this differentiation process that leads to the concept of a DAE index that is widely used in the literature. For the semi-explicit DAE systems (A. 10) that are of interest to us here, the index has the following definition. 

In principle, all the methods described above for single odes can be used for the solution of such a system, when extended suitably. In the case of explicit methods such as Euler or RK, this is very simple to implement, whereas with implicit methods such as BI or the trapezium method, there are some choices to be made. 

Essentially, only two implicit methods will be described here, but with extensions that make them more useful. They are derived from the implicit methods described for odes in Chap. 4, BI and the trapezium method. These have different names in the pde context, as will be seen. 

How one forms the approximations for the ODEs is crucial to the performance of this approach. Gear [22] and many others since showed how implicit methods convert the ODEs so that the solution method is stable and can therefore be used to solve stiff sets of equations. Implicit methods give algebraic equations that generally must be solved iteratively at each time step, as they usually involve the variables at time step k + 1 nonlinearly. 

Sincovec et al. [23] presented a very disquieting example of an initial value problem consisting of two ODEs. They showed that only one of the two state variables involved can be given an independent initial value. It was not hard to see why the problem occurs, but it was evident that such a problem could easily be hidden in a larger example. This work also proved that if an incorrect initial condition is specified and an implicit integration scheme is used, the solution will march directly to a solution that corresponds to one where a legitimate initial condition is used. The initial condition may not be one of interest, however. 

---