Implicit integration algorithms

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. 

To illustrate the difference in stability properties between explicit and implicit integration algorithms, consider again the equation used to describe valve dynamics in Section 2.2. Dropping the subscripts from equation (2.9) for clarity and generality, and setting the demanded valve travel, xj, to zero, indicating a... 

Figure 14.4 Gear and implicit Euler numerical integration algorithms.

There were no dynamic simulation issues experienced in this system. The default Implicit Euler numerical integration algorithm worked well, giving quite short simulation times (1 min of real time to simulate lOh of process time). 

G. Zhang and T. Schlick. LIN A new algorithm combining implicit integration and normal mode techniques for molecular dynamics. J. Comp. Chem., 14 1212-1233, 1993. 

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

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. 

In summary, given an initial composition c/>, and the time step At, (6.251) and (6.253) can be integrated numerically to find A, = R(c/>i i At) and A(0O At). This step will be referred to as direct integration (DI), and is implicitly assumed to yield an accurate value for 4>m given 0O- However, DI is expensive compared with interpolation. Thus, since it must be repeated at every tabulated point, DI represents the dominant computational cost in the ISAT algorithm. 

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. 

Note that, in this interpretation, must be evaluated at a position midway between X and X +i, as in an implicit midstep algorithm for numerical integration, rather than at the position obtained at the beginning of the step, as in the Ito interpretation. 

The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

Schlick and Olson recently developed such an algorithm that permits larger time steps. The implicit Euler integration scheme is combined with the Langevin dynamics formulation, which contains frictional and random... 

The DDAPLUS algorithm (Caracotsios and Stewart, 1985), updated here, is an extension of the DDASSL (Petzold, 1982) implicit integrator. DDAPLUS solves differential-algebraic equation systems of the form... 

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