Initial value problems stiffness

The index J can label quantum states of the same or different chemical species. Equation (A3.13.20) corresponds to a generally stiff initial value problem [42, 43]. In matrix notation one may write ... 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

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. 

The objective of this problem is to explore the performance of stiff and nonstiff user-oriented initial-value-problem software. Acquire the Fortran source code and the documentation for Vode from www.netlib.org. The VODE package enables the user to select either stiff or nonstiff methods. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

Initial value problems, abbreviated by the acronym IVP, can be solved quite easily, since for these problems all initial conditions are specified at only one interval endpoint for the variable. More precisely, for IVPs the value of the dependent variable(s) are given for one specific value of the independent variable such as the initial condition at one location or at one time. Simple numerical integration techniques generally suffice to solve IVPs. This is so nowadays even for stiff differential equations, since good stiff DE solvers are widely available in software form and in MATLAB. 

Moreover, the eigenvalues of the matrix A in the linearization of a nonlinear system determine the system s stiffness. We have encountered stiffness with DEs in Chapters 4 and 5 for both initial value problems (IVPs) and boundary value problems (BVPs). A formal definition of stiffness for systems of DEs is given in Section 5.1 on p. 276. 

In ref. 144 the author presents the construction of a non-standard explicit algorithm for initial-value problems. The order of the developed method is two and also is A-stable. The new proposed method is proven to be suitable for solving different kind of initial-value problems such as non-singular problems, singular problems, stiff problems and singularly perturbed problems. Some numerical experiments are considered in order to check the behaviour of the method when applied to a variety of initial-value problems. 

In ref 156 the author studies the stability properties of a family of exponentially fitted Runge-Kutta-Nystrom methods. More specifically the author investigates the P-stability which is a very important property usually required for the numerical solution of stiff oscillatory second-order initial value problems. In this paper P-stable exponentially fitted Runge-Kutta-Nystrom methods with arbitrary high order are developed. The results of this paper are proved based on a S5unmetry argument. 

Simultaneous to the graph creation, kinetic properties in each vRxn are used to create the appropriate reaction rate equations (ordinary differential equations, ODE). These properties include rate constants (e.g., Michaelis constant, Km, and maximum velocity, Vmax, for enzyme-catalyzed reactions, and k for nonenzymatic reactions), inhibitor constants, A) and modes of inhibition or allosterism. The total set of rate equations and specified initial conditions forms an initial value problem that is solved by a stiff ODE equation solver for the concentrations of all species as a function of time. The constituent transforms for the each virtual enzyme are compiled by carefully culling the literature for data on enzymes known to act on the chemicals and chemical metabolites of interest. 

In section 3.2.4, nonlinear boundary value problems were solved using shooting technique. The given boundary value problem was converted to a system of initial value problems. The unknown initial condition was obtained using an iteration and optimization procedure. This is a very robust technique and can be used to solve stiff boundary value problems. This technique is capable of predicting multiple steady states in a catalyst pellet. However, the number of iterations required for convergence can be prohibitively large for certain boundary value problems. 

ODEPACK A collection of codes for solving stiff and nonstiff systems of initial value problems. 

Cash, J. (2003). Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations. Proceedings of the Royal Society, 459, 797-815. 

If the parameters Aij, aij and Eij are all known, the initial concentrations and a temperature profile are given, the rate equations would predict the behaviour of the reaction. For very large systems a program LARKIN that integrates the, in general stiff, system of equations [27]. The initial value problems may be solved by routines like METANl [29] or SODEX [30, 31]. Both methods are based on a semi-implicit midpoint rule. 

The model for the schematic system (Fig. 3) consists of the simple ODEs (3) (or (6)), (7), (8) and (9), which form an initial value problem (IVP). In the case that pure hydrogen is used, its pressure is kept constant and the liquid-phase components are nonvolatile, the gas-phase balance equations (8)-(9) are discarded and the gas-phase concentration in eqs (3) and (6) is obtained e.g. from the ideal gas law. The initial conditions, i.e. the concentrations at t=0 are equal everywhere in the system and the IVP can be solved numerically by any stiff ODE-solver. 

We must solve this stiff, initial value problem numerically, and we use the GEAR software package (Hindmarsh, 1974). This is coded in FORTRAN, which I shall assume to the computational language of most astrophysicists. 

In ref. 164 the authors consider a new BDF fourth-order method for solving stiff initial-value problems, based on Chebyshev approximation. The authors prove that the developed method may be presented as a Runge-Kutta method having stage order four. They examine the stability properties of the method and they presented a strategy for changing the step size based on embedded pair of the Runge-Kutta schemes. 

As explained in Sect. 2.1, a full description of the time-dependent progress of a chemical reaction system requires a mechanism containing not just reactants and products but also important intermediate species. The rate of consumption of the species within the mechanism can vary over many orders of magnitude depending on the species type. Radical intermediates, for example, usually react on quicker timescales than stable molecular species. This can lead to numerical issues when attempting to solve initial value problems such as that expressed in Eq. (5.1), since the variation in timescales can lead to a stiff differential equation system which may become numerically unstable unless a small time step is used or special numerical... 

Stiff initial value problems were first encountered in the study of the motion of springs of varying stiffness, from which the problem derives its name. For linear ordinary differential equations, the stiffness of the system can be defined in terms of the stiffness ratio SR (Finlayson, 1980), given by ... 

The general approach to solving stiff equations is to use implicit methods. Historically, two chemical engineers, Curtis and Hirschfelder ([11]), proposed the first set of numerical formulas that are well-suited for stiff initial value problems by adopting ... 

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