Properties of the Nonlinear Equations

This Chapter is devoted to the study of mathematical properties of the two-dimensional non-linear shallow water equations, starting from the eigenstructure of the equations, which is discussed in terms of both conserved variables and primitive variables. Hyperbolicity of the equations in one and two space dimensions is proved and the nature of characteristic fields is established as being either linearly degenerate or genuinely non-linear. The rotational invariance of the two-dimensional equations is proved; this property is of much value when designing numerical methods for unstructured meshes. Finally, the two-dimensional steady shallow water equations are analysed, proving their hyperbolicity in the supercritical regime. A list of suggested exercises is given at the end of the Chapter. Useful background is found in Chaps.  1 and 2 .