Gear s method

For both the finite difference and collocation methods a set of coupled ordinaiy differential eqiiafions results which are integrated forward in time using the method of hnes. Various software packages implementing Gear s method are popular. 

Gear s method, 118 Gibbs (free) energy of activation (see Activation parameters Free energy) Guggenheim method, 26-27... 

MESH) equations which are solved for the whole column, decanter included and taking into account the liquid-liquid phase split. Numerical treatment of the Differential Algebraic Equation (DAE) system and discrete events handling is performed with DISCo, a numerical package for hybrid systems with a DAE solver based on Gear s method. The column technical features and operating conditions are shown in Table 4. A sequence of two operational batch steps, namely... 

So-called stiff differential equation models are particularly challenging to solve. Stiff models have dynamic behavior that encompasses a wide range of time scales. An example would be fast kinetics combined with long fluid-residence times in a chemical reactor. Gear s method is perhaps the most commonly used technique for solving these types of problems. 

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. 

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... 

This procedure had converged in 4 or 5 iterations to four significant figures for all cases tried in this study. The accuracy of the calculations depends on the time increment At because the finite difference approximations become more accurate as At gets smaller. A summary of some iteration results and a comparison between this technique and the numerical integration with Gear s method will be presented after the following discussion on the stability of the temperature equation. 

Comparison of the Computation Results. As indicated above, Gear s method was used to solve the model equations only for a fraction of the total residence time in the reactor which took 8.59 minutes of machine computation time. The same set of equations was solved by the approximate iterative technique for the same time interval in 5.8 seconds of computer time. As a comparison of the accuracy overall devolatilization Vj = Z Z v j as predicted by the two techniques are plotted on a dimensionless scale in Figure 1. The definitions for the dimensionless quantities used are ... 

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... 

The numerical Method of Lines as implemented in the routine NDSolve of the Mathematica system deals with system (32) by employing the default fourth order finite difference discretization in the spatial variable Z, and creating a much larger coupled system of ordinnary equations for the transformed dimensionless temperature evaluated on the knots of the created mesh. This resulting system is internally solved (still inside NDSolve routine) with Gear s method for stiff ODE systems. Once numerical results have been obtained and automatically interpolated by NDSolve, one can apply the inverse expression (31.b) to obtain the full dimensionless temperature field. 

We solved the above system of equations (19)—(21) by means of the method of lines and Gear s method. In addition, we performed a limited number of Brownian dynamics (BD) simulations based on particle-in-cell (PIC) method [39] with spherically symmetric concentric cells and the boundary conditions corresponding to the case (I). In these simulations, the plasma background is modelled by finite numbers of particles of two sorts representing the ion and... 

The entire equation set was solved numerically using Gear s method for stiff differential equations (Gear 1971). The initial mercury dose was administered at 100% methylmercury, administered as a bolus to the gut lumen compartment. The mass transport parameters listed in Table 2-6 were multiplied by the time-dependent compartment volumes to give the mass transport parameters used in the model equations. 

Note that Gear s method is used for this example. The following methods are available in Maple ... 

The EDA system corresponding to the model is solved by a modified version of the LSODI routine, which is based on Gear s method. The version implemented performs the solution of the EDA system concomitantly to the evaluation of the parameters sensitivities based on the decoupled direct method (4). As a matter of fact, the simulation of the system is... 

The mass balance was integrated using Gear s method, taking into account the temperature profile by linear interpolation between the experimental points. The significance of the model was judged by variance analysis [18]. The F-test revealed that the model cannot be rejected at a significance level of a = 0.05 for both the CO/O mixture and the simulated exhaust gas (fig. 3). 

Fig. 6.4. Different patterns of bursting obtained in the model for cAMP signalling. The time course and corresponding phase space trajectories are obtained by munerical integration of eqns (6.3) by means of Gear s method (NAG subroutine D02EBF) for the parameter values of fig. 6.2 from top to bottom, the values of fceand v = vI/a are (in min ) (1.8,2,4) and (0.18,0.2,0.4) (Li ct at, 1992b).

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