Stiff equations explicit methods

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

As an example, if only quasi-steady flow elements are used with volume pressure elements, a model s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)... 

As only fast reactions are responsible for the stiff charakter of the differential equation we additionally can restrict the use of the implicit method to treat the small number of fast reactions and use an explicit methods for all other reactions. The right hand side is then a sum of two parts that describe the influence of the fast reactions G and the slow reactions F... 

For stiff differential equations, an explicit method cannot be used to obtain a stable solution. To solve stiff systems, an unrealistically short step size h is required. On the other hand, the use of an implicit method requires an iterative solution of a nonlinear algebraic equation system, that is, a solution of k, from Equation A2.4. By a Taylor series development of y + /=i il i truncation after the first term, a semi-implicit Runge-Kutta method is obtained. The term k, can be calculated from [1]... 

However, a reduction of (1) to the form (2) requires a large additional numerical costs at every integration step, because this is connected with the inversion of the matrix Fy = dF/dy which generally is singular. The numerical problem appeares to be more complicated because of the stiffness of explicit equations systems in this case it is necessary to apply of special methods with conversion of the Jacobian matrix. A class of the schemes is offered [8], in which the reduction to the form (1) and the calculation of the approximate solution are carried out simultaneously. The given methods were generated by the (m, fc)-schemes [9] for solving the explicit ODE systems. 

Another way of classifying the integration techniques depends on whether or not the method is explicit, semi-implicit, or implicit. The implicit and semi-implicit methods play an important role in the numerical solution of stiff differential equations. To maintain the continuity of the section, we will first describe the explicit integration techniques in the context of one-step and multistep methods. The concept of stiffness and implicit methods are considered in a separate subsection, which also marks the end of this section. 

The Adams-Bashforth methods use information about prior points. In principle, one can form polynomials using forward points as well. Using the points Xfc+i, Xk,. .., Xfcj, to form a - - 1 polynomial generates a class of methods known as Adams-Moulton Methods. However, in these methods also calculation of yk+i requires the solution of fk+i implicitly. Implicit methods are discussed separately in a section which deals with stiff equations. One can also use a combination of an implicit method, such as an Adams-Moulton method, along with an explicit method, like an Adams-Bashforth method, to form an explicit method known as the Predictor-Corrector Method, which is discussed below. 

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. 

Numerical methods are incapable of solving nonlinear equations explicitly and the actual behavior must be approximated by a sequence of linear steps. Incremental and iterative methods are available to solve a system of nonlinear equations. In the incremental method, the response is approximated by dividing the solution into a number of linear increments and updating the stiffness at each increment. The incremental method may underestimate the nonlinear behavior and a progressive divergence from the actual response may be observed. A better approximation would be obtained by decreasing the size of the increments but for a controlled reduction in error an iterative method is required. 

Eqs. (4.19). There are two extreme possibilities for this. One is to compute these terms completely at time n St). This is the explicit method, or method of forward differences. It is easy to use, since the values of all dependent variables at the start of the time step are known. Unfortunately, the explicit approach by itself is unsuitable for dealing with coupled equations for rate processes with very different time constants. The chain reactions of combustion chemistry have just this stiff property rapid reactions reach equilibrium long before the system as a whole does. Stability of computation demands a small time step St appropriate to the rates of the faster reactions, with a corresponding prohibitive increase in the cost of the complete calculation. 

The standard discretization for the equations (9) in molecular dynamics is the (explicit) Verlet method. Stability considerations imply that the Verlet method must be applied with a step-size restriction k < e = j2jK,. Various methods have been suggested to avoid this step-size barrier. The most popular is to replace the stiff spring by a holonomic constraint, as in (4). For our first model problem, this leads to the equations d... 

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

The computational cost of the LU-SSOR scheme is comparable to that of the two-step explicit scheme. The damping properties of the error of the LU-SSOR method tend to be a bit worse when compared to explicit multistep methods, such as the simplified Runge-Kutta method. However, implicit or semi-implicit methods are preferred to solve stiff systems of equations. 

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