Stiff ordinary differential equations

the numerical solution is found and plotted until t=20  

We observe that Maple predicts negative concentration. This is because the default absolute error in dsolve numeric is only ld-6, which can be decreased to predict more accurate solutions  

Warning, cannot evaluate the solution further right of 16.734694, maxfun limit exceeded (see dsolve,maxfun for details) 

Maple s default Runge-Kutta method cannot predict the profiles after T=16.3. In addition, the program takes too long to run. This is a still problem and can be conveniently solved by using Maple s still solver. 

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Equation (4.78) is a set of nonlinear algebraic equation and may be solved using various techniques [64], Often the nonlinear differential Eq. (4.77) are solved to the steady-state condition in place of the algebraic equations using the stiff ordinary differential equation solvers described in Chapter 2 [65], See Appendix I for more information on available numerical codes. 

CONP Kee, R. J., Rupley, F. and Miller, J. A. Sandia National Laboratories, Livermore, CA. A Fortran program (conp.f) that solves the time-dependent kinetics of a homogeneous, constant pressure, adiabatic system. The program runs in conjunction with CHEMKIN and a stiff ordinary differential equation solver such as LSODE (lsode.f, Hindmarsh, A. C. LSODE and LSODI, Two Initial Value Differential Equation Solvers, ACM SIGNUM Newsletter, 15, 4, (1980)). The simplicity of the code is particularly valuable for those not familiar with CHEMKIN. 

A predictor-corrector algorithm for automatic computer-assisted integration of stiff ordinary differential equations. This procedure carries the name of its originator. ... 

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. 

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. 

Young, T. R., and Boris, J. P., A Numerical Technique for Solving Stiff Ordinary Differential Equations Associated with the Chemical Kinetics of Reactive Flow Problems, J. Phys. 

Young, CHEMEQ - A Subroutine for Solving Stiff Ordinary Differential Equations, NRL Memorandum Report 4091,... 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

In the following sections some background information on stiff ordinary differential equations will be given and the general finite difference approximations for particle temperatures will be derived. Later, the technique will be applied to coal pyrolysis in a transport reactor where the difference equations for reaction kinetics will be discussed and the calculation results will be compared with those obtained by the previously established techniques. 

A commercial stiff ordinary differential equation solver subroutine, DVOGER, is available in the IMSL Library (3). This subroutine uses Gear s method for the solution of stiff ODE s with analytic or numerical Jacobians. The pyrolysis model was solved using DVOGER and the analytical Jacobians of Eqs. (14) and (15). For a residence time of 0.0511 in dimensionless time, defined as t/t where 9... 

Models for the reacting polydispersed particles contain stiff ordinary differential equations. The stiffness is due partly to the wide range of thermal time constants of the particles and partly to the high temperature dependence of reactions like combustion and devolatilization. As an alternative to the established solution techniques based on Gear s method an iterative approach is developed which uses the finite difference representations of the differential equations. The finite differences are obtained by... 

Garcia, A. L., Bell, J. B., Crutchfield, W. Y., and Alder, B. J. J. Comp. Phys. 154(1), 134 (1999). Gear, C. W. The automatic integration of stiff ordinary differential equations, Proceedings of the IP68 Conference, North-Holland, Amsterdam (1969). 

The Selected Asymptotic Integration Method (5) has been utilized for many years at NRL for the solution of the coupled "stiff" ordinary differential equations associated with reactive flow problems. This program has been optimized for the ASC. 

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. 

The reader should note that in Eqs. (B.2)-(B.5) the spatial derivative appears on the right-hand side, and therefore it will be necessary to define a realizable high-order FVM for each case. In contrast, the source term S in Eq. (B.l) contains no spatial derivatives and hence is local in each finite-volume grid cell. In other words, with operator splitting the source term leads to a (stiff) ordinary differential equation (ODE) for each grid cell. 

Michelsen, M. L., An Efficient General Purposes Method for the Integration of Stiff Ordinary Differential Equations, AIChEJ. 22, 594-597 (1976). 

Gear CW (1969) The automatic integration of stiff ordinary differential equations. In Morrel AJH (ed) Information processing 68. North-HoUand, Amsterdam, pp 187-193... 

The MATLAB functions ode23s and ode]5s are solvers suitable for solution of stiff ordinary differential equations (see Table 5.1). 

This selection process is then iterated, beginning from an initial state of the system, as defined by species populations, to simulate a chemical evolution. A statistical ensemble is generated by repeated simulation of the chemical evolution using different sequences of random numbers in the Monte Carlo selection process. Within limits imposed by computer time restrictions, ensemble population averages and relevant statistical information can be evaluated to any desired degree of accuracy. In particular, reliable values for the first several moments of the distribution can be obtained both inexpensively and efficiently via a computer algorithm which is incredibly easy to implement (21, 22), especially in comparison to now-standard techniques foF soTving the stiff ordinary differential equations (48, 49) which may arise in the deterministic description of chemical kinetics (53). Now consider briefly the essential features of a simple chemical model which illustrates well the attributes of stochastic chemical simulations. 

Sherman, A. H. Hindmarsh, A. C. (1980). GEARS A package for the solution of sparse, stiff ordinary differential equations, Lawrence Livermore Laboratory preprint UCRL-84012. 

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