Engsig-Karup, Allan Peter3; Hesthaven, Jan S.5; Bingham, Harry B.2; Madsen, Per A.2
1 Coastal, Maritime and Structural Engineering, Department of Mechanical Engineering, Technical University of Denmark2 Department of Mechanical Engineering, Technical University of Denmark3 Department of Applied Mathematics and Computer Science, Technical University of Denmark4 Brown University5 Brown University
We present a discontinuous Galerkin finite element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one and two horizontal dimensions. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are verified for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar; and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in a future paper.
Journal of Engineering Mathematics, 2006, Vol 56, Issue 3, p. 351-370