Abstract In the previous chapter, we saw that the simple explicit numerical scheme resulted in an instability problem. We also saw that the problem could be resolved by using sufficiently short time steps. But in many situations, short time steps become exceedingly short, as can be seen, e.g., in the stability criterion (3.56). This means that we have to perform computations for a very large number of time steps to reach the final time and it is therefore tempting to look for alternatives. The most common alternative is to use an implicit scheme, which generally allows for much longer time steps.