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• Investigating one-line modelling issues • • • • # Shoreline modelling knowledge transfer project

## HR Wallingford and the University of Oxford set up a shorter knowledge transfer project to investigate several issues regarding one-line modelling.

The project was supported with government funding from the Technology Strategy Board. The main objectives of the project were to explore:

• the use of higher order explicit numerical schemes and implicit schemes for the resolution of the one-line beach response equation;
• the possibility of curvilinear coordinates and conformal mapping for one-line models.

### Resolution of the one-line beach response equation

In the one-line model the equation for conservation of sand is given by the advection-diffusion equation: where y is the shoreline position (m), D is the summation of depth of closure and berm height (m), Q is the volume rate of alongshore sediment transport (m3/sec), x is the distance along the shore (m) and t is time (sec).

### Time integration

The existing model uses an explicit Euler time integration scheme. This project looked into other explicit schemes: Adams-Bashforth 2nd order (AB2) and Runge-Kutta 4th order (RK4), concluding that Euler and RK4 give the best results in terms of stability and accuracy and therefore discarding the possibility of including AB2 in the model.

### Space integration

The numerical scheme used in the model was a forward-time forward-space first order difference method. Other methods were examined in this project: forward-time centre space finite difference (FTCS), upwind differencing (UD), three-level fully implicit scheme (3I) and Crank-Nicholson implicit scheme for the advection equation (CN).

### Conclusions

• The upwind differencing for the one-line equation is not appropriate for the case considered here, because the results become increasingly asymmetric and inaccurate. The main reason is that the diffusion component of the one-line equation dominates the advection part at low wave angles of incidence.
• The CN scheme is usually very effective when applied to the one–dimensional diffusion equation. Although this scheme allows for a bigger time step, it does not give a very consistent solution as the one-line equation is by nature an advection-diffusion equation, and so the solver should ideally comprise a combination of the diffusion and advection Crank-Nicolson schemes.
• The 3I solver allows for a bigger time step than the explicit solvers, therefore increasing stability in some instances.

### Conformal mapping/strip modelling

Boundary-fitted co-ordinate systems are useful in the numerical solution of two- and three-dimensional field problems and whilst they can be applied to the 1D case the benefit is questionable. Essentially, the method involves mapping any arbitrary-shaped domain onto a rectangle or series of rectangles. This facilitates an exact fit to irregular and curved boundaries, unlike uniform Cartesian grid schemes. The mapping process is based on the numerical solution of a pair of Poisson mapping equations using perimeter co-ordinates as boundary values. The converged solution defines the physical mesh. This mapping process can be applied to combine cross-shore models into a strip model. A working prototype of the strip model was created and plausible results achieved, which require validation.

 Authors Belén Blanco, Ian Townend, Ilektra-Georgia Apostolidou*, Prof Alistair Borthwick* Prof Paul Taylor*, *University of Oxford Keywords One-line model; numerical scheme; strip model; knowledge transfer project (KTP) Completed 2010 ### Contact Giovanni Cuomo

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