where the values on the right side are before smoothing, and 



(150) 



3> k 12k + 4 

 where 



fy,* = (1 + 4k %-i,k + >i.k + n j,k-i + U^i 3 



- k(n j-2,fe + n j + 2,k + n j,fc-2 + n j,fe + 2 ) (151) 



and k represents a weighting spline coefficient that varies from 



to °o. The influence from the surrounding points is controlled by 



the values of (k) . For the case k = 0, the equation reduces to Laplacian 



interpolation. 



To avoid numerical instability, Chen, Divcky, and Hwang (1975) imposed 

 the condition at matching points that 



^matching ~ ' ^linear + ^higher order* 

 Also, the partial derivative with respect to time was approximated by 



m = n j,fe : %i< C152) 



at At 



where n is taken at time t Q + At/2 and n at t - At/2, and 



"o,k = °- s V.* + °- 125C Vi,k + Vi.fc + V.ft-i + V.*^ (153) 



For the open boundary condition previously mentioned (Fig. 12), the 

 finite-difference equation becomes 



CAt . 



n B,fe = n S,fe " Ax lT] B,k " Vl,fc J 



where 



C- [g(d + n)] 1/2 (l + 0.5 j^-) (155) 



Listings of typical computer programs for solutions of long-wave 

 equations can be found in Brandsma, Divoky, and Hwang (1975) for linear 

 long-wave equations, and in Chen, Divoky, and Hwang (1975) for Boussinesq- 

 type equations. 



66 



