Second-Order Non-linear Methods

This chapter is devoted to the construction of second-order accurate numerical methods as applied to the shallow water equations. The schemes are non-linear, in order to circumvent Godunov’s theorem, which is concerned with the phenomenon of spurious oscillations in the vicinity of large spatial gradients, shocks in particular. For the class of flux limiter methods considered, the non-linear character of the schemes results from enforcing total variation diminishing (TVD) criteria. A second class of non-linear second-order methods is based either on TVD criteria or on non-linear spatial reconstruction of the Essentially Non-Oscillatory (ENO) type. Regarding the underlying first-order methods, two classes of methods are constructed, namely upwind methods based on the Riemann solvers studied in previous chapters and centred methods based on the FORCE framework. Numerical results from some of the methods are shown, followed by analysis and discussion on the performance of the methods tested.