Differential-algebraic equations software

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

There are many high-quality, well-documented, software packages available to solve stiff problems in this form. However, one often encounters chemically reacting flow problems that are not easily posed as standard-form ODEs. In these cases problems can often be posed easily in a more general form, called differential-algebraic equations (DAE),... 

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 equation set in this example was solved by using a differential-algebraic equation solver called gPROMS from Process Systems Enterprises (www.pse. com). It can also be solved with other software and programming languages such as FORTRAN. Example 16 is too complicated to be done on a spreadsheet. 

Macroscopic phenomena are described by systems of integro-partial differential algebraic equations (IPDAEs) that are simulated by continuum methods such as finite difference, finite volume and finite element methods ([65] and references dted therein [66, 67]). The commonality of these methods is their use of a mesh or grid over the spatial dimensions [68-71]. Such methods form the basis of many common software packages such as Fluent for simulating fluid dynamics and ABAQUS for simulating solid mechanics problems. 

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. 

In Equation-Oriented (EO) approach the software architecture is close to a solver of equations. EO is more suited for dynamic simulation since this can be modelled by a system of differential-algebraic equations (DAE) of the form ... 

Hindmarsh AC, Petzold LR (1995) Algorithms and software for ordinary differential equations and differential-algebraic equations. Part II Higher-order methods and software packages. Comput Phys 9 148-155... 

Much professional software is devoted to this problem. A diskette for sets of differential and algebraic equations with parameters to be found by this method is by Constantinides Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987). 

Figure 8 depicts our view of an ideal structure for an applications program. The boxes with the heavy borders represent those functions that are problem specific, while the light-border boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically either systems of nonlinear algebraic equations, ordinary differential equation initial or boundary value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the user must write the subroutine that defines his particular system of equations. However, that subroutine should be able to make calls to problem-independent software to return many of the components that are needed to assemble the governing equations. Specifically, such software could be called to return in-... 

A fire model is a physical or mathematical representation of burning or other processes associated with fires. Mathematical models range from relatively simple formula that can be solved analytically to extensive hybrid sets of differential and algebraic equations that must be solved numerically on a computer. Software to accomplish this is referred to as a computer fire model. 

The numerical solution is performed by the method of lines. Spatial discretization of the partial differential-equation system using finite differences on statically adapted grids leads to large systems of ordinary differential and algebraic equations. This system of coupled equations is solved by an implicit extrapolation method using the software package LIMEX [14]. The code computes species mass-fraction and temperature profiles in the gas phase, fluxes at the gas-surface interface, and surface temperature and coverage as function of time. 

Complex coupled flow and heat transfer problems can be solved using numerical techniques in which the partial differential equations are converted to a large set of coupled algebraic equations, and the algebraic equations are then solved using conventional methods developed specificaUy to be efficient on digital computers. The concept by which the numerical solution of the partial differential equations is obtained is rather straightforward, and we wiU describe it here. Actual implementation into an efficient, user-friendly computer code is difficult and tedious, however, and most users employ commercial software. 

---