Numerical Integration of Ordinary Differential Equations

As discussed in the introduction to this chapter, the solution of ordinary differential equations (ODEs) on a digital computer involves numerical integration. We will present several of the simplest and most popular numerical-integration algorithms. In Sec, 4.4.1 we will discuss explicit methods and in Sec. 4.4.2 we will briefly describe implicit algorithms. The differences between the two types and their advantages and disadvantages will be discussed. 

Let me make my own personal preference clear from the outset. I have solved literally hundreds of systems of ODEs for chemical engineering systems over my 30 years of experience, and t have found only one or two situations where the plain old simple-minded first-order Euler algorithm was not the best choice for the problem. We will show some comparisons of different types of algorithms on different problems in this chapter and the next. 

We need to study the numerical integration of only first-order ODEs. Any higher-order equations, say with Mth-order derivatives, can be reduced to N first-order ODEs. For example, suppose we have a third-order ODE  

Thus wc have three first-order ODEs to solve  

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Gear C W 1966 The numerical integration of ordinary differential equations of various orders ANL 7126... 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

W. Gautschi. Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3 381-397, 1961. 

Solving the gas dynamics expressions of Kuhl et al. (1973) requires numerical integration of ordinary differential equations. Hence, the Kuhl et al. paper was soon followed by various papers in which Kuhl s numerical exact solution was approximated by analytical expressions. 

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). 

Euler s method [15, 28] represented in Figure 7.12 is the simplest way to perform this task. Because of its simplicity it is ideally suited to demonstrate the general principles of the numerical integration of ordinary differential equations. 

Better Ceramics Through Chemistiy 11, Materials Research Society, Anaheim Gear CW (1966) The numerical integration of ordinary differential equations of various orders. In Report ANL 7126 Argotme National Laboratory... 

In this chapter, we will present a number of methods for numerical integration of ordinary differential equations. We also present orthogonal collocation method for partial differential equations as well as ordinary differential equations. A major portion of this chapter is derived from the work by the author on batch distillation [7]. 

Numerical integration of ordinary differential equations is most conveniently performed when the system consists of a set of n simultaneous first-order ordinary differential equations of the form ... 

Leonhard Euler (1707-1783) was not interested in numerical integration of ordinary differential equations, but he was interested in initial value problems. Incidentally, Euler is also responsible for what, according to Richard Feynman, is the most remarkable equation ever e + 1 = 0. 

The first type, which includes, for example, the problem of strong explosion or propagation of heat in a medium with nonlinear thermal conductivity [3], is characterized by the fact that the exponents are found from physical considerations, from the conservation laws and their dimensionality. In addition, the exponents turn out to be rational numbers. The task of the calculation is to find the dimensionless functions by integration of ordinary differential equations. After this the problem is completely solved, since the numerical constants are determined by normalizing the solution to the conserved quantity (the total energy released in these examples). 

Whether one uses Newton s or Hamilton s equations of motion, obtaining the atomic positions over time requires numerical integration. Integration of ordinary differential equations (ODE) is a well-traveled territory in numerical analysis. A number of different techniques are routinely used in MD. 

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-... 

It is useful at this point to note that the Newton forward difference formula is utilized here for the development of the numerical integration formula, while the Newton backward difference formula was previously used (in Chapter 7) for the integration of ordinary differential equations of the initial value type. 

Numerical approaches summation of intermolecular forces and integration of ordinary differential equations opposed to solving multidimensional and nonlinear partial differential equations. 

In Chapter 2 we developed a number of mathematical models that describe the dynamic operation of selected processes. Solving such models—that is, finding the output variables as functions of time for some change in the input variable(s)—requires either analytical or numerical integration of the differential equations. Sometimes considerable effort is involved in obtaining the solutions. One important class of models includes systems described by linear ordinary differential equations (ODEs). Such linear systems represent the starting point for many analysis techniques in process control. 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

Numerical integration of systems of ordinary differential equations, including... 

An implementation of this algorithm, using the sequential procedure within the MATLAB environment, was proposed by Figueroa and Romagnoli (1994). To solve step 2, the constr function from the MATLAB Optimization Toolbox has been used. The numerical integration necessary in this step has been performed via the function ode45 for the solution of ordinary differential equations. 

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