Stiff equations implicit methods

How one forms the approximations for the ODEs is crucial to the performance of this approach. Gear [22] and many others since showed how implicit methods convert the ODEs so that the solution method is stable and can therefore be used to solve stiff sets of equations. Implicit methods give algebraic equations that generally must be solved iteratively at each time step, as they usually involve the variables at time step k + 1 nonlinearly. 

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. 

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. 

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. 

Implicit methods resolve the stability problems associated with stiffness. However, because they are implicit, more work at each time step is required to solve a system of equations, which typically is nonlinear. To facilitate subsequent discussion, assume that a... 

If the integration is taking an interminable amount of time, it is possible that you have defined a stiff system of equations (see Appendix F). In that case, change to an implicit method (for stiff equations, hence the s-designation) ... 

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... 

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. 

The numerical methods that are better for solving stiff problems perform more computational work in each step and can take larger steps, such as the implicit methods. There are also solutions that were developed to solve stiff systems of equations among the solutions applied to problems of moderate size appears the Differential Algrebraic System Solver (DASSL). In this solution, the derivatives are approximated by the backward formnla, and the nonlinear system resulting in each step is solved by the Newton s method. 

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]... 

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. 

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 ... 

The very short time constant in the second equation requires a very small h to follow both y and z in time. Problems of instability can easily arise. To solve stiff problems, the most typical approach is to use an implicit method, which is known to exhibit excellent stability properties (recall the backward Euler method). Special software packages are available for solving stiff systems. Fortunately, many of the straightforward chemical engineering problems enconntered in practice do not yield stiff systems, but when difficulties arise, stiffness might well be the culprit. 

M72 Solution of stiff differential equations semi-implicit Runge-Kutta method with backsteps Rosenbrock-Gottwa1d-Wanner 7200 7416... 

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