Stiff Equations and Implicit Methods

The concept of stiffness is related to stability and can be understood in the context of the following simple problem of a batch reactor. The first order reaction of a batch reactor is given below. 

If we apply Euler s method to this problem, recursively, one can arrive at the following approximation to the exact solution (equation 2.16). 

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  

Comparison of Euler s method with exact solution 

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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