DASSL

Petzold, L. R. A Description of DASSL A Differential-Algebraic System Solver, Sandia National Laboratory Report SAND82-8637 also in Stepleman, R. S. et al., eds. IMACS Trans, on Scientific Computing, vol. 1, pp. 65-68. 

This equation must be solved for y The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL(Ref. 224). 

The boundary conditions were used to obtain special forms of these equations at the boundary nodes. The complete pelletizer model contained a total of 207 differential and algebraic equations which were solved simultaneously. The differential/algebraic program, DASSL, developed at Sandia National Laboratories 2., .) was used. The solution procedure is outlined in Figure 5. 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

High-level DAE software (e.g., Dassl) makes a time-step selection based on an estimate of the local truncation error, which depends on the difference between a predictor and a corrector step [13,46]. If the difference is too great, the time step is reduced. In the limit of At 0, the predictor is just the initial condition. For the simple linear problem illustrated here, the corrector will always converge to the correct solution y2 = 1, independent of the time step. However, if the initial condition is y2 1, then there is simply no time step for which the predictor and corrector values will be sufficiently close, and the error estimate will always fail. Based on this simple problem, it may seem like a straightforward task to build software that identifies and avoids the problem, and there is current research on the subject [13], The problem is that in highly nonlinear, coupled, problems the inconsistent initial conditions can be extremely difficult to identify and fix in a general way. 

Develop two method-of-lines simulations to solve this problem. In the first, formulate the problem as standard-form ordinary differential equations, y7 = ff(f, y). In the second, formulate the problem in differential-algebraic (DAE) form, 0 = g(t, y, y ). Standard-form stiff, ordinary-differential-equation (ODE) solvers are readily avalaible. DAE solvers are less readily available, but Dassl is a good choice. The Fortran source code for Dassl is available at http //wwwjietlib.org. 

Write the appropriate driver routine to integrate the equations using either the DASSL [46] (DAE) or VODE (ODE) [49] software. You will need to iteratively solve for the laboratory-referenced molar fluxes N], N2 using a multidimensional root-finding routine, such as MNEWT / FDJAC [319], a Newton-Raphson scheme. 

Problems like plug flow can be posed as standard-form ODEs, but it is much more convenient to pose them as DAEs. Other situations, such as boundary-layer flow (Chapters 7 and 17) are difficult to pose as standard-form ODEs, but a DAE formulation works well. The Dassl family of software [46] is designed for solving DAEs and is used extensively in the Chemkin software. 

Importantly, recognize that the sensitivity problem is a linear equation for the sensitivity coefficients regardless of whether the original problem is linear or nonlinear. Once the solution to the underlying problem is determined, the sensitivity coefficients can be computed efficiently, exploiting the inherent linearity [57,102,110,232,321], There is recent sensitivity software by Petzold that builds on the DASSL family of codes [258],... 

After acquiring the FORTRAN source code and documentation for Dassl (or other differential-algebraic solver) from www.netlib.org, write a simulation program to solve this problem. 

Taken together, the system of equations represents a set of stiff ordinary differential equations, which can be solved numerically. Because more than one dependent-variable derivative can appear in a single equation (e.g., the momentum equation has velocity and pressure derivatives), it is usually more convenient to use differential-algebraic equation (DAE) software (e.g., Dassl) for the solution rather than standard-form ODE software. 

The numerical solution is accomplished with a method-of-lines approach, using a control-volume spatial discretization. The time integration can be done using Dassl, which implements an implicit, variable-order, variable-step, method based on the BDF method [46],... 

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. 

The dynamic behavior of the reactor can be simulated by solving Eqs. (1)—(6). The differential-algebraic solver DASSL [14] is used to give the solution of these equations. The initial conditions for MA, MB, Me, Mo used in all simulation studies are 12, 12, 0, and Okmol, respectively. The initial values of both reactor and jacket temperature are set to 20 °C. Other process parameter values used in the reactor models are listed in Table 1. 

L.R. Petzold, A description of DASSL a differential/algebraic system solver, SAND82-8637, Sandia National Laboratories, 1982. 

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. 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

Petzold, L. R., A description of DASSL A differential/algebraic system solver, in Scientific Computing, eds. R. S. Stepleman et al., North-Holland, Amsterdam, (1983). 

With previously published kinetic constants [10,11], presented in Table 1, the model was solved for two different initial conditions. The former one (type I) presumes the existence of a reactants and products profile at t=0, whereas the latter (type II) considers that the reactor is empty at t=0. The resultant differential-algebraic system of equations was solved by backward finite differences formula with variable step, implemented in the DASSL code [12,13], The numerical convergence was assured by increasing the number of finite elements until no further modification in the model simulations was obtained. Hence, the number of elements was gradually increased fi om 20 to 100. Nevertheless, no significant diference was observed, i. e., all product yields were obtained with errors smaller than 10 . ... 

The mathematical model of a MAT reactor, considering a 12 lump model, has been discretized using a finite element method in the direction of gas flow. The resulting system of differential-algebraic equations (DAEs) has been solved by an appropriate computer code (DASSL). 

L.R.Petzold, DASSL Code, version 1989, Computing and Mathematics Research Division,... 

Only numerical solutions of the VERSE model can be obtained [65]. The partial differential equations are discretized by application of the method of orthogonal collocation on fixed finite elements. Equation 16.59 is divided into 50 or 60 elements, each with four interior collocation points. Legendre polynomials are used for each element. For Eq. 16.62, only one element is required. It is described by a Jacobi polynomial with two interior collocation points. The resulting set of ordinary differential equations, with their initial and boundary conditions and the chemical equations, are solved using a differential algebraic system solver (DASSL) [65,66]. 

---