A non‐iterative algorithm based on Richardson's extrapolation. Application to groundwater flow modelling

In this report, a promising third‐order algorithm based on Richardson's extrapolation and Crank–Nicholson's method is studied. The algorithm, having a wide scope for applications, was initially intended to be used for time integration on non‐linear advection–diffusion problems. Then stability, low oscillations and low additional diffusion become important numerical properties. It is shown that this method, using three single steps for extrapolation, is very close to be unconditionally stable. Under the previous requirements it behaves essentially as well as Crank–Nicholson's scheme but with a better performance, at least in relation to spurious oscillations, and perhaps not so good for dispersion properties. In addition the method is third‐order accurate, and provides useful information related to time integration error to be used for time step control. These features indicate that the algorithm is a good candidate to efficiently solve problems with significant variations in time. Additionally, its non‐iterative version saves important amounts of CPU time, specially for applied 3D modelling. To illustrate the good performance of the algorithm we include several tests comparing it with Crank–Nicholson's scheme, and a 2D application to variably saturated groundwater flow modelling. Copyright © 2005 John Wiley & Sons, Ltd.