Ordinary differential equations, numerical computational methods

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... 

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. 

In [170] the authors discussed the numerical solution of Ordinary Differential Equations (ODEs) by using two approached the well known BDF formulae and the Piecewise-Linearized Methods. In the case of BDF method a Chord-Shamanskii iteration procedure is used for computing the nonlinear system which is produced when the BDF formula is applied. In the case of Piecewise-Linearized Methods the computation of the numerical solution at each time step is obtained using a block-oriented method based on diagonal Pade approximation. 

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)]. 

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. 

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. 

To compute unsteady flows, the time derivative terms in the governing equations need to be discretized. The major difference in the space and time co-ordinates lies in the direction of influence. In unsteady flows, there is no backward influence. The governing equations for unsteady flows are, therefore, parabolic in time. Therefore, essentially all the numerical methods advance in time, in a step-by-step or marching approach. These methods are very similar to those applied for initial value problems (IVPs) of ordinary differential equations. In this section, some of the methods widely used in the context of the finite volume method are discussed. 

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. 

The GRM using the generalized Maxwell-Stefan equations has no closed-form solutions. Numerical solutions were calculated using a computer program based on an implementation of the method of orthogonal collocation on finite elements [29,62,63]. The set of discretized ordinary differential equations was solved with the Adams-Moulton method, implemented in the VODE procedure [64]. The relative and absolute errors of the numerical calculations were 1 x 10 and 1 x 10 , respectively. 

A second option is to use existing packages for numerical methods. The software libraries by IMSL and NAG have a wide variety of state-of-the-art numerical integrators. These libraries are well documented, reliable, and flexible, and can be found at most university computing centers or networks. The packages Matlab, Mathematica, and Maple are more interactive and also have programs for solving ordinary differential equations. 

Similarity Criteria and Reduced Equations The partial differential equations (Eqs. 6.95-6.97) are not amenable to solution except by numerical methods utilizing high-speed computers. Considerable simplifications can result, as in the case of the flat plate, if these equations are reduced to ordinary differential equations through the similarity concept where the dependent variables f, w, and 1 are assumed functions of -q alone. Equations 6.95-6.97 become, for t, = constant... 

In the previous section we solved linear ordinary differential equations analytically, obtaining general solutions in terms of the parameters in the equations. Numerical methods can also be used to obtain solutions, using a computer. In Chapter 1 we looked at the dynamic responses of several processes by using numerical integration methods (Euler integration-see Table 1.2). 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

The set of equations (10.5-22) is quite readily handled by the numerical method of orthogonal collocation. Basically, the coupled partial differential equations (eq. 10.5-22) are discretized in the sense that the spatial domain r is discretized into N collocation points, and the governing equation is valid at these points. In this way, the coupled partial differential equations will become coupled ordinary differential equations in terms of concentrations at those points. These resulting coupled ODEs are function of time and are solved by any standard ODE solver. Details of the orthogonal collocation analysis are given in Appendix 10.5, and a computer code ADSORB3A is provided with this book for the readers to learn interactively and explore the simulation of this model. 

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