Numerical Solution of Ordinary Differential Equations

APPENDIX 2 NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is 

A value of At is selected, and values for A a, A b,. .. are estimated by evaluating the functions 3bA, 3bg. In Euler s method, this evaluation is done at the initial point (a0, b0, , t0) so that the estimate for A a is just At3bA(ao, bo,. .., to) — At(3bA)0. In fourth-order Runge-Kutta, the evaluation is done at four points and the estimates for A a, A b,. .. are based on weighted averages of the dbA, 3bB. at these four points  

Example 2.14 Use fourth-order Runge-Kutta integration to solve the following set of ODEs  

Solution The coding is left to the reader, but if you really need a worked example of the Runge-Kutta integration, check out Example 6.4. The following are detailed results for At = 1.0, which means that only one step was taken to reach the answer. 

Although not recommended for practical use, the classical Euler extrapolation is a convenient example to illustrate the basic ideas and problems of numerical methods. Given a point (tj y1) of the numerical solution and a step size h, the explicit Euler method is based on the approximation (yi+1 - /( i+l 4 dy/dt to extrapolate the solution 

As seen from Fig. 5.2, reducing the step size this estimation. 

While in the first step the deviation from the exact solution stems only from approximating the solution curve by its tangent line, in further steps we calculate the slope at the current approximation y1 instead of the unknown true value (), thereby introducing additional errors. The solution of (5.2) is given by (5.3), and the total error Ei = y(t ) - y for this simple equation is 

The first term in (5.9) is the local truncation or step error that occurs in a single step and does not take into account the use of y instead of y( t j ). The second term shows the propagation of the error E j. It is of primary importance to keep the effect of E decreasing in latter steps, resulting in the stability of the method. In this simple example the 

stability can be achieved only at sufficiently small step sizes. Such steps decrease also the truncation error, but increase the required computational effort. Therefore, a common goal of all numerical methods is to provide stability and relatively small truncation errors at a reasonably large step size (refs. 1-2). 

For a batch reactor, the input and output terms are zero therefore, the material balance simplifies to 

Assuming that reaction (5,1) takes place in the liquid phase with negligible change in volume, Eq. (5,3) written for each component of the reaction will have the form 

Consider the growth of a microorganism, say a yeast, in a continuous fermentor of the type shown in Fig. 5.1. The volume of the liquid in the fermentor is V. The flow rate of nutrients into the fermentor is F, and the flow rate of products out of the fermentor is The material balance for the cells X is 

If we make the assumption that the fermentor is perfectly mixed, that is, the concentrations at every point in the fermentor are the same, then 

Further assumptions are made that the flow rates in and out of the fermentor are identical, and that the rates of cell formation and substrate utilization are given by 

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Shampine S 1994 Numerical Solutions of Ordinary Differential Equations (New York Chapman and Hall)... 

Numerical Solution of Ordinary Differential Equations as Initial... 

Lapidus, L., and J. Seinfeld. Numerical Solution of Ordinary Differential Equations, Academic, New York (1971). 

NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS... 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

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 ). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

Figure 3.6 Numerical solution of ordinary differential equations sketch of the four steps of the Runge-Kutta method to the order four giving the n+1 th estimate y(n+1) from the nth estimate y ". ...

In this chapter, the numerical solution of ordinary differential equations (odes) will be described. There is a direct connection between this area and that of partial differential equations (pdes), as noted in, for example [558]. The ode field is large but here we restrict ourselves to those techniques that appear again in the pde field. Readers wishing greater depth than is presented here can find it in the great number of texts on the subject, such as the classics by Lapidus k Seinfeld [351], Gear [264] or Jain [314] there is a very clear chapter in Gerald [266]. 

References General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equa-... 

T. E. Simos, Numerical Solution of Ordinary Differential Equations with Periodical Solution, Doctoral Dissertation, National Technical University of Athens, Greece, 1990 (in Greek). 

A brief survey of the different classes of ODE methods that are commonly applied solving the governing equations of fluid mechanics, is given in the following. The elementary notation and the basic properties of these ODE discretizations are briefly mentioned. Analysis of these methods can be found in numerous textbooks on the numerical solution of ordinary differential equations and will not be repeated here [156, 66, 170, 158, 134, 93, 28]. 

Shampine LF (1994) Numerical solution of ordinary differential equations. Chapman Hall, New York... 

Byrne, G.D., and A.C. Hindmarsh, A polyalgorithm for the numerical solution of ordinary differential equations. ACM Trans. Math Software 1, 71, 1975. 

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