The Shallow Water Equations -- 3.1 Introduction

3 Finite volume method for the numerical solution of the
flat-bottom case

3.1 Introduction

3.1 Introduction

The most obvious - and perhaps the most successful - choice for the numerical treatment of our system of equations is the finite volume method. This is because the numerical scheme incorporates the notion of "weak solutions" which include discontinuities. As such, a finite volume method is dealing with certain volume averages of the quantities and with fluxes across cell boundaries rather than with a point-to-point discretization of the differential operators (finite difference method). Hence, the starting point for the discretization is an integral (or weak) formulation of the equations. As in [7] we first integrate our basic one-dimensional, non-viscous relation over a spatial domain (a,b),

and then in time over (n*Dt,(n+1)*Dt):

In the above relation - which is exact - f corresponds to the time average of the flux during the period of integration. The implications are important: if we suppose some discretization of our spatial domain into finite volumes V_i and define the following spatial average between cell boundaries (i-1/2) and (i+1/2),

we obtain from the equation:

It becomes clear that the temporal variation of the cell-averaged values is due to the time integral of the cell-flux difference. This statement of physical conservation can be considered as the foundation for numerical finite volume methods. The main part of the remaining task is to find physically meaningful numerical approximations to the fluxes f.

markus.uhlmann AT ciemat.es

3.1 Introduction

Contents

Index