Digital Object Identifier (DOI) 10.1007/s00791-002-0090-8 Comput Visual Sci 5: 85–94 (2002) Computing and Visualization in Science  Springer-Verlag 2002 Regular article Algebraic factorizations for 3D non-hydrostatic free surface flows ∗ Paola Causin, Edie Miglio, Fausto Saleri Dipartimento di Matematica “F.Brioschi”, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Received: 2 April 2001 / Accepted: 10 February 2002 Communicated by: G. Wittum Abstract. The high amount of computer resources required to simulate complex free surface flows has prompted for devel- oping fractional step schemes capable of reducing the com- putational effort. These schemes are borrowed from a wider family of methods originally devised for the incompressible Navier–Stokes equations. An alternative approach is to per- form an algebraic splitting on the coefficient matrix of the linear system resulting from the discretized problem, ending up with the successive solution of sub-problems of smaller size. The resulting schemes are shown in different cases to be the algebraic counterpart of the standard fractional step formulations. This algebraic procedure was again originally devised in the context of incompressible Navier–Stokes sys- tem, but we believe it is far more general: in this paper it is indeed extended to the more involved 3D free surface flow model. The inexact block factorization technique is applied to the coefficient matrix arising from the problem at hand and two significant choices for the approximation are discussed and numerically tested. 1 Introduction Free surface flows are encountered in many natural phenom- ena, from tidal currents, to wide water basins, to river courses. Full 3D numerical simulations require great computa- tional efforts; therefore, in the past only 1D and 2D models have been considered. Nowadays, these models are well es- tabilished, both in terms of a sound mathematical formula- tion and of a robust numerical implementation (see for ex- ample [1] and [4]). Nonetheless, there exist many practical situations where a more accurate physical description of the problem is re- quired in order to correctly represent all the significant phe- nomena. With this aim, in the last decade 3D models for the shallow water equations (shortly 3D-SWE) have been ∗ This work has been carried out using the computational resources avail- able at CINECA. developed: they are thought of as an extension of two di- mensional models and are tailored to describe selected three- dimensional properties of the flow, with a computational cost not severely exceeding the cost of pure 2D models. The 3D-SWE model is derived from the 3D incompressible Navier–Stokes equations by integrating the continuity equa- tion along the vertical direction, while retaining a full-three dimensional velocity field. A common further simplification hypothesis assumes that the vertical accelerations of the fluid are negligible, yielding an hydrostatic pressure field that is related to the height of the surface. A dimensional analysis of the equations demon- strates that the hydrostatic approximation is correct provided that the ratio of the vertical to horizontal motion scales (in term of velocities and lengths) is small. The physical observa- tions attest that currents over areas with steep bathymetry gra- dients or flows around obstacles are typical problems where the hydrostatic hypothesis has to be removed and the fully 3D Free Surface model must be considered. The algebraic system obtained in this last case after perform- ing the discretization in time and space is very demanding to be solved. Therefore a fractional step approach represents a suitable choice in order to reduce the computational effort. A convenient and physically motivated adaptation of the NS pressure-velocity splitting to the 3D-SWE case consists in decoupling the problem into the successive solution of an hy- drostatic sub-problem and an hydrodynamic correction [14]. Precisely, upon splitting the pressure into hydrostatic and hy- drodynamic contributions, the direct connection between the former term and the height of the fluid is taken advantage of and a first system is solved for velocities and elevation; then, in a following step, an hydrodynamic correction is obtained by enforcing the final velocity field to be divergence-free. The aim of this work is to perform a somewhat parallel procedure using algebraic splitting schemes, i.e. to generalize to the non-hydrostatic free surface problem the methodolo- gies proposed and analyzed in [17, 18] after [16] for the NS problem. Following this idea, the fractional step scheme illus- trated above will be cast within the mathematical framework