An optimizing implicit difference scheme based on proper orthogonal decomposition for the two‐dimensional unsaturated soil water flow equation

SUMMARY An optimizing reduced implicit difference scheme (IDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for the two‐dimensional unsaturated soil water flow equation is presented. An ensemble of snapshots is compiled from the transient solutions derived from the usual IDS for a two‐dimensional unsaturated flow equation. Then, optimal orthogonal bases are reconstructed by implementing SVD and POD techniques for the ensemble of snapshots. Combining POD with a Galerkin projection approach, a new lower dimensional and highly accurate IDS for the two‐dimensional unsaturated flow equation is obtained. Error estimates between the true solution, the usual IDS solution, and the reduced IDS solution based on POD basis are derived. Finally, it is shown by means of a numerical example using the technology of local refined grids that the computational load is greatly diminished by using the reduced IDS. Also, the error between the POD approximate solution and the usual IDS solution is proved to be consistent with the derived theoretical results. Thus, both feasibility and efficiency of the POD method are validated. Copyright © 2011 John Wiley & Sons, Ltd.