Of ODE solvers

This book describes and gives Fortran subroutines for a wide variety of ODE solvers. More to the point, it gives numerical recipes for practically anything you will ever need to compute. Volume 2 is also available online. It discusses Fortran 90 in the context of parallel computing. C, Pascal, and Basic versions of Volume 1 can be purchased. 

Hindmarsh, A.C. (1983) ODEPACK-A systematized collection of ODE solvers, in Scientific Computing (eds R. Stepleman et al), IMACS/North Holland Publishing Company, pp. 55-64. 

A.C. Hindmarsh. ODEPACK, a Systemized Collection of ODE Solvers, pages 55-64. North-Holland, Amsterdam, 1983. 

A C Hindmarsh, ODEPACK A systematized collecton of ODE solvers, Elsevier, Amsterdam 1983... 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE Solvers," in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64... 

Hindmarsh, AC, "ODEPACK, a systematized collection of ode solvers in scientific computing", RS Stepleman et al. ( s.), north-holland, amsterdam, 1983, pp. 55-64. 

In this chapter the basics of numerical integration methods for explicit ODEs will be presented. The goal is to discuss the fundamental ideas, eg. order, stability, step size selection etc.. We will focus also on the main components of modern implementations of ODE solvers and point out special modifications which can exploit the structure of equations of motion of mechanical systems. 

General purpose methods adapt the step size to the solution of the problem, and they also adapt the order of the method, which allows both high accuracy and stability. The stability of the methods generally becomes worse as the order increases. The maximum order can be specified for stability reasons. A simple numerical experiment can be used to show how the selection of ODE solver affects a solution. Consider Example 6.6, which becomes stiff for large a values. 

Explain what is meant by stiff differential equations, and what category of ODE solvers is suitable for stiff systems. 

A list of ODE solvers (and of other routines that act on functions) is returned by help funfun, and a documentation window is opened by doc funfun. The two main routines of interest are ode45, an explicit single-step integrator, and odelBs, an implicit multistep integrator that works well for stiff systems. We demonstrate the use of ode45 and odelBs for a simple batch reactor with the two elementary reactions A -I- B C and C -I- B D... 

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