Spectral/hp Discontinuous Galerkin Methods for Computational Hydraulics
We present the concept of spectral/hp element methods, i.e. finite element methods of arbitrarily (high) polynomial order, and apply the methods to computational hydraulics. More specifically, we use the spectral/hp version of the discontinuous Galerkin method to depth-integrated wave equations, describing the propagation and evolution of surface gravity waves. We are particularly interested in solving Boussinesq-type equations for simulation of dispersive waves in the time-domain. The major objective for this study is to explore if the spectral/hp element approach can generate computationally competitive models for coastal and hydraulic engineering. Even with today's powerful computers, immense computational times preclude time-domain modelling of dispersive waves in large-scale coastal areas.
First, the nonlinear shallow water equations are discretized on unstructured triangles of size h using an orthogonal modal expansion of arbitrary order p in space. The numerical fluxes are evaluated using an approximate Riemann solver. We then extend the model to include dispersive terms of lowest-order, corresponding to the classical Boussinesq equations. In solving the Boussinesq equations we propose the concept of a 'scalar method'. In the scalar method we rewrite the coupled momentum equations as a scalar 'wave continuity equation', used as an intermediate step in solving the momentum equations. The size of the resulting sparse matrix system is thereby significantly reduced. Even if the original variables need to be recovered in a subsequent step, the scalar method is shown to be more CPU efficient, as well as requiring less storage, than directly solving the coupled momentum equations. Finally, we include additional dispersive terms, corresponding to a set of enhanced Boussinesq-type equations.
The models are shown to have optimal convergence and accuracy of O(hp+1) for smooth problems. Thus the models exhibit the expected exponential convergence with regard to polynomial order. More importantly, compared to low-order schemes, the efficiency of the models are demonstrated to increase with (i) increasing polynomial order, (ii) increasing time of integration and (iii) increasing relative depth. Thus, the spectral/hp element method appears to offer potentially significant savings in computational time for large-scale long-time dispersive wave simulations.
surface gravity waves
shallow water equations
discontinuous Galerkin method