A high-order Godunov-type scheme based on MUSCL variable extrapolation and slope limiters is presented for the resolution of 2D free-surface flow equations. In order to apply a finite volume technique of integration over body-fitted grids, the construction of an approximate Jacobian (Roe type) of the normal flux function is proposed. This procedure allows conservative upwind discretization of the equations for arbitrary cell shapes. The main advantage of the model stems from the adaptability of the grid to the geometry of the problem and the subsequent ability to produce correct results near the boundaries.
Verification of the technique is made by comparison with analytical solutions and very good agreement is found. Three cases of rapidly varying two-dimensional flows are presented to show the efficiency and stability of this method, which contains no terms depending on adjustable parameters. It can be considered well suited for computation of rather complex free-surface two-dimensional problems.
KEY WORDS Free-surface flow Two-dimensional modelling Finite volumes MUSCL approach Upwind differencing
1. Introduction
In recent years many advances have been made in the study of hyperbolic partial differential equations and in the theory of one-dimensional difference operators applied to hyperbolic partial differential equations describing fluid flows.' Much is known about numerical techniques for solving conservation laws, i.e. equations of the form Solutions of such equations produce discontinuities in general (unlessfis a linear function of u) and numerical methods have been developed that can handle these non-linear features.
The numerical resolution of systems of conservation laws stems from the work of Lax and Wendroff (1960)," who formally expressed the importance of the conservative discretizations to capture automatically the discontinuities present in the solution. In the 1970s the concepts of artificial viscosity and modified equation were introduced and the numerical schemes such as Lax-Wendroff or MacCormack became popular.
At the end of the 1970s the original ideas of Godunov about the incorporation of the physical reality to the solution of simple problems were reconsidered. At the same time van Leer analysed 027 1-209 1/93/0604&9-17$13.50 ( i 1993 by John Wiley & Sons, Ltd.