Implicit methods and stability

All the methods presented so far, e.g. the Euler and the Runge-Kutta methods, are examples of explicit methods, as the numerical solution aty +i has an explicit formula. Explicit methods, however, have problems with stability, and there are certain stability constraints that prevent the explicit methods from taking very large time steps. Stability analysis can be used to show that the explicit Euler method is conditionally stable, i.e. the step size has to be chosen sufficiently small to ensure stability. This conditional stability, i.e. the existence of a critical step size beyond which numerical instabilities manifest, is typical for all explicit methods. In contrast, the implicit methods have much better stability properties. Let us introduce the implicit backward Euler method. 

Consider the problem of prediciting the concentration of a reactant in an isothermal batch reactor, assuming a first-order reaction, where the reaction rate, r, is given by the product of the reaction rate constant, k, and the concentration of the reactant, y  

In this equation, we have introduced the time constant of the system, which is the inverse of the rate constant k, i.e. x = /k. This simple problem involves a linear differential equation that allows us to investigate how the implicit and the explicit methods behave as the step size is modified. Recall that the explicit Euler method is given by y +i = y +, y ). This means that 

Consequently, the implicit method will be stable even for the poor choice of time step h. However, in order to achieve an appropriate accuracy, the step size h has to be chosen reasonably small. This simple example differs from most practical applications in one very important aspect. In this case, it was simple, using algebra, to rewrite the original implicit formula as an explicit one for evaluation. This is usually not the case in practical applications. In such cases, the implicit method is more complex to use, and it involves solving fory +i, using indirect means. 

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