A one-dimensional benchmark for the propagation of Poincaré waves

Several numerical methods are employed to solve the linear shallow-water equations describing the propagation of Poincaré waves within a one-dimensional finite domain. An analytical solution to the problem, set off by a discontinuous steplike elevation, is known and allows for assessing the accuracy and robustness of each method and in particular their ability to capture the traveling discontinuities without generating spurious oscillations. The following methods are implemented: the method of characteristics, the Galerkin finite-element method (FEM) and the discontinuous Galerkin FEM with two different ways of computing the numerical fluxes.