Numerical methods algebraic equations

As mentioned in Chapter 2, the numerical solution of the systems of algebraic equations is based on the general categories of direct or iterative procedures. In the finite element modelling of polymer processing problems the most frequently used methods are the direet methods. 

As mentioned earlier, overall accuracy of finite element computations is directly detennined by the accuracy of the method employed to obtain the numerical solution of the global system of algebraic equations. In practical simulations, therefore, computational errors which are liable to affect the solution of global stiffness equations should be carefully analysed. 

M. J. D. Powell, numerical Methods forNon-Einear Algebraic Equations, Gordon and Breach, New York, 1970. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

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). 

V. The auxiliary equation is normally an algebraic equation rather than an ODE. In chemical engineering problems, it will usually be an equation of state, such as the ideal gas law. In any case, the set of ODEs can be integrated numerically starting with known initial conditions, and V can be calculated and updated as necessary. Using Euler s method, V is determined at each time step... 

The numerical methods in this book can be applied to all components in the system, even inerts. When the reaction rates are formulated using Equation (2.8), the solutions automatically account for the stoichiometry of the reaction. We have not always followed this approach. For example, several of the examples have ignored product concentrations when they do not affect reaction rates and when they are easily found from the amount of reactants consumed. Also, some of the analytical solutions have used stoichiometry directly to ease the algebra. This section formalizes the use of stoichiometric constraints. 

Usually the finite difference method or the grid method is aimed at numerical solution of various problems in mathematical physics. Under such an approach the solution of partial differential equations amounts to solving systems of algebraic equations. 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

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)]. 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. 

Digital simulation is a powerful tool for solving the equations describing chemical engineering systems. The principal difficulties are two (1) solution of simultaneous nonlinear algebraic equations (usually done by some iterative method), and (2) numerical integration of ordinary differential equations (using discrete finite-difference equations to approximate continuous differential equations). 

The above nonlinear feedforward controller equations were found analytically. In more complex systems, analytical methods become too complex, and numerical techniques must be used to find the required nonlinear changes in manipulated variables. The nonlinear steadystate changes can be found by using the nonlinear algebraic equations describing the process. The dynamic portion can often be approximated by linearizing around various steadystates. 

We have found that dynamics can be more conveniently handled in the Russian transfer-function language than in the English ODE language. However, the manipulation of the algebraic equations becomes more and more difficult as the system becomes more complex and higher in order, if the system is th-order, an Afth-order polynomial in s must be factored into its N roots. For N greater than 2, we usually abandon analytical methods and turn to numerical... 

Brenan, K. E., and Petzold, L. R., The numerical solution of higher index differential/algebraic equations by implicit Runge-Kutta methods," UCRL-95905, preprint, Lawrence Livermore National Laboratories, Livermore, California (1987). 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

I will return to this diagram near the end of the chapter, particularly to amplify the meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1. For now, I will illustrate the bootstrapping technique for improving phases, map, and model with an analogy the method of successive approximations for solving a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems defy such analytic solutions but are amenable to numerical methods. The method of successive approximations has much in common with the iterative process that extracts a protein model from diffraction data. 

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