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Numerical methods for the non-linear shallow water equations

Tsima Alexandra

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Summary

In the present postgraduate thesis we study the shallow water equations and several finite-volume numerical methods, of first and second order, as well as centred methods that are used to solve these equations. In the first chapters of this thesis we consider the derivation of the shallow water equations in one dimension from the conservation laws, the hyperbolic character of the equations as well as their eigenvalues and eigenvectors. Of main interest is the understanding of primitive and conservative variables as in the numerical methods the right choice of the variables is vital, for the right evaluation of the shock waves. The second part of this thesis is a comparative study of several numerical methods. We solve appropriately chosen problems, each of them has its own difficulty and we study if they can solve the problems sufficiently. In TVD methods we use several limiters and observe if they exist or not differences in the numerical results of each problem and how important is the choice of a limiter. Finally, we consider the shallow water equations in two space dimensions, which are an expansion of the shallow water equations in one-dimension. We study the wave propagation phenomena associated with the sudden collapse of an idealized two dimensional circular dam and the bore reflection patterns that occur when a bore reflects from a solid vertical wall at an angle to the bore propagation direction.

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