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ical methods to verify the validity of the solution and resulting computational code. 

 The following results were produced by a Fortran code derived from the code written 

 and generously provided by Takumi Ohyama that was used to compute the water 

 surface plots given in (Ohyama et al. 1995a). The code was expanded to compute 

 the full kinematics. 



A. 4.1 Richardson Extrapolation 



Fenton (1985a) presented a variation of Richardson extrapolation that can be used 

 for numerical verification of wave theory code. Richardson extrapolation (also known 

 as extrapolation to the limit) is a method traditionally used to increase the accuracy 

 of finite difference computations (Salvadori and Baron 1961; Roach 1976). In Fenton's 

 adaptation, the method is used to simultaneously check all the terms in the solution, 

 and to provide an estimate of the order of the accuracy of the result. 



The form of Ohyama et al.'s solution solves the field equation (Laplace) and the 

 bottom boundary condition exactly. It is expected to satisfy the free surface boundary 

 conditions to the order it is applied. The following method calculates the order of 

 accuracy of the free surface boundary conditions. Classical Richardson extrapolation 

 is used to evaluate the error of a solution at a given point in time and space. By 

 taking advantage of the periodicity of wave motion, Fenton's variation can be used 

 to evaluate the solution throughout a complete wave. 



Single Wave 



The solution for a single wave is periodic and symmetric about the crest, and thus 

 the error in each of the boundary conditions can be expanded in a single sided Fourier 

 series: 



k=oo 



E{e) = J2 Ck{e)e''''^'^ (A.7) 



