Computation of the Motion of Long Water Waves 



where variables without superscript are known at the n time level. 



Second, the x-momentum Eq. (4) is used to obtain a difference 

 equation for u values at the n+i level. Central space-differences 

 and time -splitting are used about the point (i»j,n+|^); this leads to 

 a difference equation that is implicit. Now we gather the u"* terms 

 from the j*^ row on the left-hand side of the equation and all other 

 terms on the right-hand side. The heart of our procedure is to use 

 uP, vP and t)P values in lieu of the u"*', v"*' and r)"*'' terms which 

 appear on the right-hand side of the equation, these terms being 

 mostly in the j-1 and j+1 rows. The result, when rearranged for 

 computation is, for each j, 



Au"V- ^ B"*' + Cu"*'. = E, i = 3,4,...,M-2 (19) 



where 



A = ^ ^ - ^2 



4A '} 2A' 



Si +^(4^- dijd..,.) 



2 



B = 1 + -^ 

 3A 



2 

 ^ At . ^ fdn .. \ 



^= 4A Sj ■" 7X^1-3^- ^iJ^i*IJ; 



2A 



E = U-- - -r-r U-(U- ,• - U^ .•) 



IJ 4A IJ i + lj i-lj' 



At , , P P . 



+ ^ ( - d: li U; . , : + 2d; :U: ; " d- ,. U- ,. 



2A ^ ' ^ ^ ^ J '" J 



X 1 /^ ^ A P A P 



+ 4 Vai4lj*l Vj^lj + I - di_,j^,Vj.,^^, - dj^,. _, v^^ ,._, 



p 

 + dj.ij., Vj.ij., - dj^ij^iVj^ij^i + dj_,.^, Vj.ij^i 



+ dj^,j.,v.^,._, - d._,.,,Vj_,..^))- 



157 



