Numerical Solution Scheme 



167. If all information is available to use Equation 1 (shoreline 

 change equation), Equation 2 (longshore sand transport rate equation), and 

 Equation 14 (wave breaking criterion) , the response of the shoreline to wave 

 action can be calculated. Under certain simplified conditions, closed-form 

 mathematical solutions of Equation 1 can be found (see, for example, Larson, 

 Hanson, and Kraus 1987), but in order to describe realistic structure and 

 shoreline configurations, including waves that vary alongshore and with time, 

 Equation 1 must be solved numerically. In a numerical solution procedure, the 

 distance alongshore is divided into cells of a certain width (called the grid 

 spacing), and the duration of the simulation is similarly divided into small 

 elements (called the time step). If the grid spacing and time step are small, 

 solutions of the governing partial differential equation (Equation 1) can be 

 accurately calculated by numerical solution of the finite-difference equation. 



Numerical and physical accuracy 



168. Referring to Figure 6 and the shoreline change equation 

 (Equation 1), the change in position of the shoreline can be mathematically 

 written as 



a At AQ , . 



Ay " " (D B + D c ) Ax" (23) 



in which AQ is the difference in longshore sand transport rates at the walls 

 of the cell. In arriving at Equation 23, the contribution to Ay by line 

 sources and sinks q was omitted for simplicity. Equation 23 indicates that 

 the change in shoreline position Ay is directly proportional to At and 

 inversely proportional to Ax (actually, Ay is inversely proportional to 

 (Ax) 2 , as described below) . 



169. Numerical accuracy refers to the degree to which the numerical 

 scheme provides an accurate solution to the partial differential equation 

 (Equation 1). Physical accuracy refers to the degree to which Equation 1 and 

 the associated input data represent the actually occurring processes. 

 Physical accuracy depends on the quality of the input data and the degree to 



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