Least-squares finite-element method for shallow-water equations with source terms

Numerical solution of shallow-water equations(SWE) has been a challenging task because of its nonlinearhyperbolic nature, admitting discontinuous solution, andthe need to satisfy the C-property. The presence of sourceterms in momentum equations, such as the bottom slope andfriction of bed, compounds the difficulties further. In thispaper, a least-squares finite-element method for the spacediscretization and θ-method for the time integration is developedfor the 2D non-conservative SWE including the sourceterms. Advantages of the method include: the source termscan be approximated easily with interpolation functions, noupwind scheme is needed, as well as the resulting systemequations is symmetric and positive-definite, therefore, canbe solved efficiently with the conjugate gradient method. Themethod is applied to steady and unsteady flows, subcriti-caland transcritical flow over a bump, 1D and 2D circulardam-break, wave past a circular cylinder, as well as wavepast a hump. Computed results show good C-property, conservationproperty and compare well with exact solutions andother numerical results for flows with weak and mild gradientchanges, but lead to inaccurate predictions for flows withstrong gradient changes and discontinuities.