Ordinary differential equations implicit methods

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. 

A few examples will be demonstrated in the following section. Each feature was incorporated in the software because it has been found necessary or useful in some modelling project. All the modelling tasks that previously required tailor made solutions in each project can now be solved in a unified manner in the ModEst environment. ModEst is able to deal with explicit algebraic, implicit algebraic (systems of nonlinear equations) and ordinary differential equations. As the standard way to handle PDE systems, the Numerical Method of Lines, which transforms a PDE system to a number of ODE components is used. In addition, any model with a solver provided may be dealt with as an algebraic system. The basic numerical tools are contained in the well tested public domain software (Bias, Linpack, Eispack, LSODE). 

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

Systems 3.1 and 3.2 contain N number of equations. The algebraic equation system can be solved with the Newton-Raphson method [1] and the ordinary differential equations with the semi-implicit Runge-Kutta method, Michelen s semi-implicit Runge-Kutta method [2], or, alternatively, with the Rosenbrock-Wanner semi-implicit Runge-Kutta method [2],... 

Levykin, A.I., Novikov, E.A. A study of (m, k)-methods of the order 3 for implicit systems of ordinary differential equations. Novosibirsk, Preprint Computing Center SB RAS 882 (1990)... 

By this means, the partial differential equation is transferred into an ordinary differential equation in the discrete cosine space. A semi-implicit method is used to trade-off the stability, computing time, and accuracy [39,40]. In order to remove the shortcomings with the small time-step size associated with the exphcit Euler scheme to achieve convergence, the linear fourth-order operators can be treated implicitly while the nonlinear terms can be treated explicitly. The resulting first-order semi-implicit Fourier scheme is ... 

There are explicit and implicit numerical methods for the finite-difference version of Eqs. 9.27 and 9.28. There are also approximate numerical methods in which the radial derivatives are replaced by the functions obtained by differentiating assumed trial functions for the radial profiles, essentially transforming the equations into ordinary differential equations in z. The cell model (Hlavacek and Votruba, 1977) can also be used for the solution. 

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

To apply Laplace transforms to practical problems of integro-differential equations, we must develop a formal procedure for operating on such functions. We consider the ordinary first derivative, leading to the general case of nth order derivatives. As we shall see, this procedure will require initial conditions on the function and its successive derivatives. This means of course that only initial value problems can be treated by Laplace transforms, a serious shortcoming of the method. It is also implicit that the derivative functions considered are valid only for continuous functions. If the function sustains a perturbation, such as a step change, within the range of independent variable considered, then the transform of the derivatives must be modified to account for this, as we will show later. 

The numerical solution is performed by the method of lines. Spatial discretization of the partial differential-equation system using finite differences on statically adapted grids leads to large systems of ordinary differential and algebraic equations. This system of coupled equations is solved by an implicit extrapolation method using the software package LIMEX [14]. The code computes species mass-fraction and temperature profiles in the gas phase, fluxes at the gas-surface interface, and surface temperature and coverage as function of time. 

In ref. 145 the authors develop two families of explicit and implicit BDF methods (Backward Differentiation Methods), for the accurate integration of differential equations with solutions any linear combinations of exponential of matrices, products of the exponentials by polynomials and products of those matrices by ordinary polynomials. More specifically, the authors study the numerical solution of the problem ... 

A similar equation can be set up for the pressure drop. The combined model now contains K+1 coupled parabolic partial differential equations and one ordinary first order differential equation. They are solved by discretization in the radial direction by use of the orthogonal collocation method, and integration of the resulting set of coupled first order differential equations by use of a semi-implicit Runge-Kutta method. With this model and the used solution method, one can now concentrate on the effective transport properties given in PeH, Pcm and the wall heat transfer coefficient, with the latter being the most important parameter for design. 

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