Simulations

All files are in (avi) format.

Linear free surface waves

Test case: Linear Harmonic Waves in a wave basin with periodic boundaries.

Numerical Scheme: Space-time discontinuous Galerkin Scheme.

Description: Space-time discontinuous Galerkin Scheme accommodates deforming grids which has advantage to model nonlinear waves satisfying the geometric conservation laws to the discrete level. To test the space-time DG scheme on deforming grid, we randomly move the mesh nodes w.r.t time and simulate the linear free surface waves which propagate undisturbed by the mesh movement. This demonstrates that the method is conservative on deforming grids and gives confidence to simulate nonlinear free surface waves. Furthermore, it is a good nontrivial test case for testing the numerical scheme for deforming grids.

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Nonlinear free surface waves

Test case: Standing waves in a wave basin with solid boundaries.

Numerical Scheme: Space-time discontinuous Galerkin Scheme for nonlinear free surface waves.

Description: Standing waves are exact solutions of linear free surface water wave equations. However, they can be tested with the nonlinear numerical scheme from low to high amplitude. Under low amplitudes, the numerical results must match with exact solutions and at high amplitudes, some nonlinear behavior must be seen.

Standing waves for one time period
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Standing waves for ten time periods
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Test case: Linear harmonic waves with periodic boundaries.

Numerical Scheme: Space-time discontinuous Galerkin Scheme for nonlinear free surface waves.

Description: Linear harmonic waves are exact solutions of linear free surface water wave equations. However, they can be tested with the nonlinear numerical scheme from low to high amplitude. Under low amplitudes, the numerical results must match with exact solutions and at high amplitudes, some nonlinear behavior must be seen. We actually observe steepening of waves due to nonlinearities. As the waves steepen, we stop our numerical algorithm as it is not designed to handle overturning waves.

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