Backward differentiation

Vode, solves stiff systems of ordinary differential equations (ODE) using backward differentiation techniques [49]. It implements rigorous control of local truncation errors by automatic time-step selection. It delivers computational efficiency by automatically varying the integration order. 

Dassl, solves stiff systems of differential-algebraic equations (DAE) using backward differentiation techniques [13,46]. The solution of coupled parabolic partial differential equations (PDE) by techniques like the method of lines is often formulated as a system of DAEs. It automatically controls integration errors and stability by varying time steps and method order. 

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. 

The model can be solved by discretizing the axial coordinate S using finite differences. Considering k = 1... N points along the column, where k = 1 corresponds to the bottom (gas inlet) and k = N corresponds to the top location (liquid inlet), the differential equations are transformed into a set of 7 x N algebraic equations. For example, after applying a backward differentiation scheme, Eq. (12.66) becomes ... 

The flow velocity within the porous panicle is estimated by the continuity equation assuming constant outflow of the produced gases from the particle. The one dimensional conservation equations are discretized in space by a finite volume approach and a backward differentiation formula is applied to time integration. 

Given the initial conditions (concentrations of the 22 chemical species at t = 0), the concentrations of the chemical species with time are found by numerically solving the set of the 22 stiff ordinary differential equations (ODE). An ordinary differential equation system solver, EPISODE (17) is used. The method chosen for the numerical solution of the system includes variable step size, variable-order backward differentiation, and a chord or semistationary Newton method with an internally computed finite difference approximation to the Jacobian equation. 

Before we illustrate how to apply Eq. 6.2.40, a comment on numerical differentiation is in order. To achieve higher accuracy of the derivatives, we use second-order differentiation relations (see Appendix C). Therefore, when the data points are equally spaced, we calculate the slope of each point, except the two endpoints, using the central differentiation equation. For the first point, we use the backward differentiation equation, and, for the last point, we use the forward differentiation equation. When the points are not equally spaced, we use the central differentiation equation for the midpoint between any two adjacent data points. Hence, for n data points on species concentrations, we obtain n — 1 derivative values for the midpoints concentrations. 

A number of other backward differentiation methods and numerical routines for their use can be found in Gear (1971). These algorithms have been rewritten and incorporated in the package ODEPACK (Hindmarsh 1983). 

The set of PDEs was written in dimensionless ffarm. The spatial discretization is performed using the Keller box scheme and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations. The resulting system is solved using a backward differentiation formula method. 

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). 

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]). 

These moment equations are typically integrated by linear multistep methods, such as the Adams method and the backward differentiation formula (Petzold, 1983). The right-hand sides of Eqs. (10.6), (10.7) are directly related to the selected reaction scheme. As an example. Scheme 10.1 shows the main reaction steps for a simplified chain-growth radical polymerization. 

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... 

Based on the above assumptions, the model equations are shown in Table 4. The mass balance equations at the pellet and crystal level are based in the double linear driving model equations or bidisperse model[30]. The solution of the set of parabolic partial differential equations showed in Table 4 was performed using the method of lines. The spatial coordinate was discretized using the method of orthogonal collocation in finite elements. For each element 2 internal collocation points were used and the basis polynomial were calculated using the shifted Jacobi polynomials with weighting function W x) = (a = Q,p=G) hat has equidistant roots inside each element [31]. The set of discretized ordinary differential equations are then solved with DASPK solver [32] which is based on backward differentiation formulas. 

MA28 or MA48 based on LU decomposition BDNLSOL block decomposition nonlinear solver SPARSE Newton-type method without block decomposition DASOLV based on variable time step/variable order backward differentiation formula. 

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