1.4 Present Approach

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1.4 Present Approach

The theory of characteristics allows to construct solutions to a number of elementary problems including discontinuities (such as the hydraulic jump). However, the analysis is restricted to simple cases such that a numerical solution is desirable. This is our present objective.

In the present note we will first develop in detail the analytical solution to the Riemann (discontinous initial-value) problem for the shallow water equations in the flat-bottom case. In this we will follow the general technique laid out by Smoller [5]. Besides its direct importance for dam-break flows the Riemann problem exhibits the essential physics encountered in more complex cases while keeping the geometric configuration simple and tractable. As such it has become a reference test case of choice for validating numerical methods. Moreover, the solution to the very Riemann problem has found its way into a good part of the modern numerical schemes conceived for the shallow water equations (and in fact for most hyperbolic systems).

In section 3 we will then present several more recent numerical approaches applicable to our system (1.1). We will especially discuss the difficulties associated with the numerical treatment of the source terms in the non-uniform bottom case which deprive the system of its conservation property.

1.4 Present Approach

Contents

Index