Stiff integration algorithm

Tackling stiffness in process simulations the properties of a stiff integration algorithm... 

A common method of solution is the Newton-Raphson method, already described in connection with a stiff integration algorithm in Section 2.7, equations (2.71) to (2.74). The equations above are in the form... 

A systematic stepwise method for numerical integration of a rate expression [indeed, of any differential equation y = f(x,y) with an initial value y(Xo) = Vo] to determine the time evolution of the rate process. See also Numerical Computer Methods Numerical Integration Stiffness Gear Algorithm... 

NUMERICAL COMPUTER METHODS NUMERICAL INTEGRATION STIFFNESS GEAR ALGORITHM... 

The characteristics of the system presented here requires a simulation tool which supports the decomposition into subsystems. With the parameters we used the system is stiff [6]. Algorithms for the numerical integration of stiff differential equations [5] and numerical libraries for solving nonlinear implicit equations like eq. (2.7) must be available. The simulation tool MATLAB/SIMULINK was used because it fulfils these requirements [11],[16]. Object-oriented visual programming helps to represent the model as shown in Fig. 2.3 and 2.4. The costly numerical solution of eq. (2.7) has been performed before the simulation and the results has been stored in a data field. 

With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

The preceding equations form a set of algebraic and ordinary differential equations which were integrated numerically using the Gear algorithm (21) because of their nonlinearity and stiffness. The computation time on the CRAY X-MP supercomputer for a typical case in this paper was about 5 min. Further details on the numerical implementation of the algorithm are provided in (Richards, J. R. et al. J. ApdI. Polv. Sci.. in press). 

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

The fundamental difficulty in solving DEs explicitly via finite formulas is tied to the fact that antiderivatives are known for only very few functions / M —> M. One can always differentiate (via the product, quotient, or chain rule) an explicitly given function f(x) quite easily, but finding an antiderivative function F with F x) = f(x) is impossible for all except very few functions /. Numerical approximations of antiderivatives can, however, be found in the form of a table of values (rather than a functional expression) numerically by a multitude of integration methods such as collected in the ode... m file suite inside MATLAB. Some of these numerical methods have been used for several centuries, while the algorithms for stiff DEs are just a few decades old. These codes are... 

A finite element program with four-noded isoparametric elements was used to solve the above governing equations and boundary conditions. Algorithm 10 presents the scheme used to evaluate the element stiffness matrices and force vectors using numerical integration. 

Algorithm 10 Computing the element stiffness matrix and force vector for a four-noded isoparametric element by numerical integration... 

The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

S is nothing other than the stiffness ratio of the system. The number of steps for Euler s explicit algorithm increases with the stiffness of the system. This conclusion is quite general or, in other words, explicit algorithms are not suitable for integrating stiff systems. 

The algorithm consists of the judicious application of one of two integration formulas to each equation in the system and the choice of formula is based on the time constant for each equation evaluated at the beginning of each chemical time step. Species with time constants too small are treated by the stiff method and the remaining species are treated by a classical second order method. The algorithm is characterized by a high degree of stability, moderate accuracy and low overhead which are very desirable features when applied to reactive flow calculations. 

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