Simulating Changing Basin Morphology Using Shoreline Tracking and the Water Wave Equations Jared Barber Program in Applied Mathematics, University of Arizona Collaborators: Juan M. Restrepo, James M. Hyman, Brad Weir

Applications Shoreline Erosion Shoreline Erosion Wave Runup Wave Runup Wave Setup and Setdown Wave Setup and Setdown Flooding and Draining Flooding and Draining Water/Ice dynamics (w/land) Water/Ice dynamics (w/land)

Numerical Goals Solving SWWE Solving SWWE Encompasses all numerical difficulties Encompasses all numerical difficulties Simplified sediment dynamics model Simplified sediment dynamics model Multiply-connected domains Multiply-connected domains Efficiency and flexibility for 1, 2, and 3-d Efficiency and flexibility for 1, 2, and 3-d Bottom topography erosion-wet and dry sand Bottom topography erosion-wet and dry sand Conservative Solver Conservative Solver Convergent Convergent

Background-Others Leveque (2006, unpublished)-Riemann Solver FVM Leveque (2006, unpublished)-Riemann Solver FVM Brocchini et al (2001)-Piston Problem at shore Brocchini et al (2001)-Piston Problem at shore Y. Uchiyama (2004)-POM Y. Uchiyama (2004)-POM Prasad and Svendsen (2003)-SHORECIRC Prasad and Svendsen (2003)-SHORECIRC Toro (1992)-WAF method Toro (1992)-WAF method

Background-Shallow Water Wave Equations u Assume: Assume: SWWE: SWWE: Conservative Eqns Conservative Eqns Add in ρ Add in ρ Mass and Momentum Mass and Momentum

Lagrangian Shoreline Tracking and Changing h(x,t) First Equation in Lagrangian Coords for shoreline First Equation in Lagrangian Coords for shoreline Use other shoreline eqns: Use other shoreline eqns: Add in eqn for h (p=0, numerically satisfactory): Add in eqn for h (p=0, numerically satisfactory):

Integration Process

Find

Integration Process Find ODE integrate u s eqn.

Integration Process Find ODE integrate u s eqn. Integrate SWWE with FVM-LF

Integration Process Find ODE integrate u s eqn. Integrate SWWE with FVM-LF Integrate h(x,t) eqn

Integration Process Find ODE integrate u s eqn. Integrate SWWE with FVM-LF Integrate h(x,t) eqn Make necessary mesh adjustments

Results-CG LF: 32 cells-Works Well-Copies Real thing

Results-Islands Works Well-Adaptive/Robust

Results-Changing h Works Well-Able to incorporate changing h

Improvements/Future Work Converges to Analytical Solutions Converges to Analytical Solutions Highly adaptive/Robust-appearing and disappearing islands Highly adaptive/Robust-appearing and disappearing islands Efficient and energy and mass conserving Efficient and energy and mass conserving Boundary discontinuities: Implement dry sand dynamics Boundary discontinuities: Implement dry sand dynamics Moving Mesh-ODE and FVM Integration, Interpolation Moving Mesh-ODE and FVM Integration, Interpolation Generalize to 2-d Generalize to 2-d Able to incorporate other phenomena (h) Able to incorporate other phenomena (h)

Thanks Ocean Sciences Conference Ocean Sciences Conference Juan Restrepo, Brad Weir - Collaborators Juan Restrepo, Brad Weir - Collaborators