Differential-Algebraic Equations dassl

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)]. 

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. 

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. 

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). 

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, A description of DASSL A differential algebraic equation system solver. 

Petzold has provided a numerical package, dassl, to compute solutions to implicit differential, and differential-algebraic equations. The main difference between using dassl and 1 sode is the form of the user-supplied function defining the model A second difference is that the user must supply xo as well as xo- The reader can consult the software for Example 8.1 at www.che,wisc.edu/ jbraw/chemreacfun to see the details of how to use dassl. 

Simultaneous solution of the so-called differential-algebraic equation (DAE) set requires coupling of the ODE and algebraic equation solvers, the latter which are not discussed here, but can be found in detail elsewhere [ 1 ]. Description of a DAE set and its solution in the context of a one-dimensional (ID) heterogeneous packed-bed reactor model for autothermal conversion of methane to hydrogen is available in the literature [7]. It is also worth noting that packages such as DASSL and DAEPACK are also available for the solution of coupled DAE sets. 

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. 

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 . ... 

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]. 

The dimensionless equations are solved by discretizing the first and second order derivatives, thus, reducing the system of differential equations into a system of algebraic equations that can be solved by using DASSL or ACSL numerical packages. 

This means that the mathematical model to be deduced will consist of two explicit differential equations including resistor currents and a set of coupled implicit algebraic equations for the outputs of the resistors and the input of the controlled source. As the model is linear, these algebraic equations could be solved symbolically turning the DAE system into an explicit ODE system. Alternatively, the DAE system could be directly formulated in the Scilab script language and evaluated by the DASSL solver. 

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