Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow

An automatic time stepping scheme with embedded error control is developed and applied to the moisture-based Richards equation. The algorithm is based on the first-order backward Euler scheme, and uses a numerical estimate of the local truncation error and an efficient time step selector to control the temporal accuracy of the integration.

Local extrapolation, equivalent to the use of an unconditionally stable Thomas–Gladwell algorithm, achieves second-order temporal accuracy at minimal additional costs. The time stepping algorithm also provides accurate initial estimates for the iterative non-linear solver. Numerical tests confirm the ability of the scheme to automatically optimize the time step size to match a user prescribed temporal error tolerance.

An important merit of the proposed method is its conceptual and computational simplicity. It can be directly incorporated into existing or new software based on the backward Euler scheme (currently prevalent in subsurface hydrologic modelling), and markedly improves their performance compared with simple fixed or heuristic time step selection.

The generality of the approach also makes possible its use for solving PDEs in other engineering applications, where strong non-linearity, stability or implementation considerations favour a simple and robust low-order method, or where there is a legacy of backward Euler codes in current use. Copyright © 2001 John Wiley & Sons, Ltd.