Ordinary differential equations, numerical

Bader, G. Deuflhard, P. A semi-implicit midpoint rule for stiff systems of ordinary differential equations. Numer. Math. 1983, 41, 373-398. 

Batch and bio process modeling often results in a set of ordinary differential equations. Numerical solution techniques for solving these differential equations normally involve approximating the differential equations by difference equations that are solved in a step-by-step marching fashion. 

Gear C W 1971 Numerical Initial Value Problems in Ordinary Differential Equations (Englewood Cliffs, NJ Prentice-Hall)... 

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

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

L.R. Petzold, L.O. Jay, and J. Yen. Numerical solution of highly oscillatory ordinary differential equations. Acta Numerica, pages 437-484, 1997. 

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. 

Numerical Solution of Ordinary Differential Equations as Initial... 

Gear, G. W. Numerical Initial Value Frohlems in Ordinary Differential Equations, Prentice Hall, Englewood CBffs, NJ (1971). 

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... 

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

The ordinary differential equations for f and C now form a fifth-order system which can be solved using a standard NAG library routine. The results are shown in Fig. 10.73. This figure also shows the numerical results for concentration obtained using a full numerical approach, and there is reasonable agreement between the two. 

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. 

The steady state TMB model equations are obtained from the transient TMB model equations by setting the time derivatives equal to zero in Equations (25) and (26). The steady state TMB model was solved numerically by using the COLNEW software [29]. This package solves a general class of mixed-order systems of boundary value ordinary differential equations and is a modification of the COLSYS package developed by Ascher et al. [30, 31]. 

In the general case of a piston flow reactor, one must solve a fairly small set of simultaneous, ordinary differential equations. The minimum set (of one) arises for a single, isothermal reaction. In principle, one extra equation must be added for each additional reaction. In practice, numerical solutions are somewhat easier to implement if a separate equation is written for each reactive component. This ensures that the stoichiometry is correct and keeps the physics and chemistry of the problem rather more transparent than when the reaction coordinate method is used to obtain the smallest possible set of differential... 

A marching-ahead solution to a parabolic partial differential equation is conceptually straightforward and directly analogous to the marching-ahead method we have used for solving ordinary differential equations. The difficulties associated with the numerical solution are the familiar ones of accuracy and stability. 

Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

The preceding equations form a set of algebraic and ordinary differential equations which were integrated numerically using the Gear algorithm (21) because of their nonlinearity and stiffness. The computation time on the CRAY X-MP supercomputer for a typical case in this paper was about 5 min. Further details on the numerical implementation of the algorithm are provided in (Richards, J. R. et al. J. ApdI. Polv. Sci.. in press). 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

Lambert JD. Numerical methods for ordinary differential equations. New York, John Wiley, 1993. 

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. 

In this chapter we concentrate on dynamic, distributed systems described by partial differential equations. Under certain conditions, some of these systems, particularly those described by linear PDEs, have analytical solutions. If such a solution does exist and the unknown parameters appear in the solution expression, the estimation problem can often be reduced to that for systems described by algebraic equations. However, most of the time, an analytical solution cannot be found and the PDEs have to be solved numerically. This case is of interest here. Our general approach is to convert the partial differential equations (PDEs) to a set of ordinary differential equations (ODEs) and then employ the techniques presented in Chapter 6 taking into consideration the high dimensionality of the problem. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

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