Backward Differentiation Formulas BDF

The BDF method is ascribed to Curtiss k Hirschfelder [188], who described it in 1952, although Bickley [88] had essentially, albeit briefly, mentioned it already in 1941. Considering Fig. 4.3, the method can be seen as a multipoint extension of BI the derivative y is formed by using a number k of points from y. k i to yn+1, but referred to the new point yn+. This implies a backward derivative, with formulas of the form y n(n) as in Appendix A, Table A.l. For example, using the three points shown in Fig. 4.3 (in other words, k = 3), the table yields the formula 

It turns out that although the BDF schemes achieve higher and higher orders as k increases, the solution begins to oscillate (certainly when the method is adapted to pdes) at about k = 5 and becomes unstable for k 7. As applied to diffusion simulations by the Feldberg school [236,402], a value of 5 is normally used. The choice of this parameter is discussed in a later chapter. 

Note that the parameter k as defined here, being the number of time points used for the backward difference, which is the convention in electrochemistry since [402], differs from the usage in computer science, where k refers to the number of intervals ( levels ) between these points, and is thus smaller by one. It is the electrochemical usage that is adhered to in this book. 

It turns out that although the BDF schemes achieve higher and higher orders as k increases, the solution begins to oscillate (certainly when the method is adapted 

---

Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful. 

Thus, after j iterations, when T = jST, the current is presented as belonging to (j — O.S)ST. The only argument for this is that it seems to work, providing more accurate current values for systems like the Cottrell experiment. However, there is no formal justification for the trick. This is a fudge and should not be used. The trick is also applied in the case of backward differentiation formula (BDF) (see Sect. 1.3.9) and there it is fully justified, as will be seen later. 

A characteristic of DAEs besides their form is their differentiation index [32]. For a definition and an example see Appendix C. It is an indicator for the problems to be encountered with the numerical solution of a set of DAEs. Systems of index > 1 are usually called higher index DAEs and the higher the index the more severe numerical difficulties can be. As the mathematical description of problems in various disciplines often leads to DAE system, they have been a research subject for more than two decades. A large body of publications and a number software programs for their numerical solution have emerged. DAE systems of index 1 can be safely numerically computed by means of the backward differentiation formula (BDF) [33, 34] implemented in solvers such as the well known and widely used DASSL code [35]. 

The RD model consists of sets of algebraic and differential equations, which are obtained from the mass, energy and momentum balances performed on each tray, reboiler, condenser, reflux drum and PI controller instances. Additionally, algebraic expressions are included to account for constitutive relations and to estimate physical properties of the components, plate hydraulics and column sizing. Moreover, initial values are included for each state variable. A detailed description of the mathematical model can be found in appendix A. The model is implemented in gPROMS /gOPT and solved using for the DAE a variable time step/variable order Backward Differentiation Formulae (BDF). 

Chapter 9 gives a review on how multibody systems can be modelled by means of multibond graphs. A major contribution of the chapter is a procedure that provides a minimum number of break variables in multibond graphs with ZCPs. For the state variables and these break variables (also called semi-state variables) a DAE system can be formulated that can be solved by means of the backward differentiation formula (BDF) method implemented in the widely used DASSL code. The approach is illustrated by means of a multibond graph with ZCPs of the planar physical pendulum example. 

Several algorithms have been developed to obtain the set of equations. The result is a set of differential-algebraic equations (DAEs) solved using a backward differential formulae (BDF) numerical method. 

The backward differential formulas (BDF) have also been used to solve stiff systems of equations, with orders ranging from one to five. For example, solving... 

---