Adaptive Time Step Methods

In practical computations, one seeks to achieve a desired accuracy with the minimum computational effort. For a given method, this requires finding the largest possible value of the time step Δt. In the previous chapters we kept the step size constant through the solution interval, but this is rarely the most efficient approach, since the error depends on the characteristics of the solution in addition to the step size. In smooth regions, larger time steps can be used without introducing significant error, while smaller time steps are needed in regions where the solution has rapid variations. This chapter extends the Runge-Kutta methods from the previous chapters to methods that select the time step automatically to control the error in the solution.