Numerical solution of ODEs

The two methods are BDF and extrapolation. Both methods are used for the numerical solution of odes and are described in Chap. 4. The extension to the solution of pdes is most easily understood if the pde is semidiscretised that is, if we only discretise the right-hand side of the diffusion equation, thus producing a set of odes. This is the Method of Lines or MOL. Once we have such a set, as seen in (8.9), the methods for systems of odes can be applied, after adding boundary conditions. 

In numerical solutions of ODE-IVP, the solution y at the point t" is represented by y". The simplest method is Euler s method, which is obtained by writing a difference expression for the derivative ... 

Gear86] Gear C. W. (1986) Maintaining solution invariants in the numerical solution of ODEs. SIAM J. Sci. Stat. Comput. 7(3) 734-743. 

Sha86] Shampine L. F. (1986) Conservation laws and the numerical solution of ODEs. Comp, and Maths, with Appls., Part B. 12. 

When it comes to the numerical solution of ODEs, everyone begins with Euler s method because it is easy to understand and simple to program. Even though its low accuracy keeps it from being widely used for solving ODEs, it gives us an idea of the basic concept of numerical solution for a differential equation simply and clearly. Let us consider a first-order differential equation ... 

In case of start-up flow from the rest state, a numerical solution of ODE set (11.4) and (11.5a)-(11.5c) should be obtained using the following initial conditions ... 

Stability of Numerical Solutions of ODEs It can be instructive to study the simple ODE-IVP... 

N concentrations. Years ago, this fact was useful since numerical solutions to ODEs required substantial computer time. They can now be solved in literally the blink of an eye, and there is little incentive to reduce dimensionality in sets of ODEs. However, the theory used to reduce dimensionality also gives global stoichiometric equations that can be useful. We will therefore present it briefly. 

The numerical solution of Equations (9.14) and (9.24) is more complicated than the solution of the first-order ODEs that govern piston flow or of the first-order ODEs that result from applying the method of lines to PDEs. The reason for the complication is the second derivative in the axial direction, Sajdz. ... 

The next two steps after the development of a mathematical process model and before its implementation to "real life" applications, are to handle the numerical solution of the model s ode s and to estimate some unknown parameters. The computer program which handles the numerical solution of the present model has been written in a very general way. After inputing concentrations, flowrate data and reaction operating conditions, the user has the options to select from a variety of different modes of reactor operation (batch, semi-batch, single continuous, continuous train, CSTR-tube) or reactor startup conditions (seeded, unseeded, full or half-full of water or emulsion recipe and empty). Then, IMSL subroutine DCEAR handles the numerical integration of the ode s. Parameter estimation of the only two unknown parameters e and Dw has been described and is further discussed in (32). 

Alternately the numerical solution of Eqs (1) and (2) can be found directly by software ODE, without going through Eq (5). 

In general, the numerical solution of PDEs is much more difficult to automate than the solution of initial-value ODEs. The best method to be used is very dependent on the problem being solved. 

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 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]. 

A note is in order here on errors in the numerical solution of an ode. There are (regarding errors in a certain light) two kinds of errors. One is the local error, being the error added by a single step. The solution is always carried forward to a final point in t, using a number N of steps, and at that point we have a final, or global error. Unfortunately, this is always of a lower order than the local error. 

All the techniques described above can also be applied to the numerical solution of systems of odes, and here we are getting closer to what happens when we solve pdes, because in effect, one reduces them to ode systems when discretising them. 

Over the past ten years the numerical simulation of the behavior of complex reaction systems has become a fairly routine procedure, and has been widely used in many areas of chemistry, [l] The most intensive application has been in environmental, atmospheric, and combustion science, where mechanisms often consisting of several hundred reactions are involved. Both deterministic (numerical solution of mass-action differential equations) and stochastic (Monte-Carlo) methods have been used. The former approach is by far the most popular, having been made possible by the development of efficient algorithms for the solution of the "stiff" ODE problem. Edelson has briefly reviewed these developments in a symposium volume which includes several papers on the mathematical techniques and their application. [2]... 

In ref. 142 the authors are studied the Numerov-type ODE solvers for the numerical solution of second-order initial value problems. They present a powerful and efficient symbolic code in MATHEMATICA for the derivation of their order conditions and principal truncation error terms. They also present the relative tree theory for such order conditions along with the elements of combinatorial mathematics, partitions of integer numbers and computer algebra which are the basis of the implementation of the S5unbolic code. We must that one of the authors is an expert on this specific field. 

The simultaneous numerical solution of the ODE system was performed using Matlab (The MathWorks). The kinetic and model parameters are summarized in Table 1,... 

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. 

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]. 

The numerical solution of Equations 9.18 and 9.28 is more complicated than the solution of the first-order ODEs that govern piston flow or of the first-order ODEs... 

FIGURE 25.9 Numerical solution of the stiff ODE example using the explicit Euler method and At = 0.001. Also shown is the true solution. 

In the example above, a term in the solution important for only 10 3 min dictated the choice of the timestep. If a step larger than 103 min is used, the solution is unstable. Similar problems almost always accompany the numerical solution of systems of ODEs describing a set of chemical reactions. These systems are characterized by timescales that vary over several orders of magnitude and are characterized as numerically stiff. For example, the differential equations (25.107) have timescales on the order of 10 3 min and 10 min. Stiffness can be defined rigorously for a system of linear ODEs that can be expressed in the vector form... 

The numerical solution of these five coupled ODEs is illustrated in Figure 2-3. Solution (b)... 

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