Computation of the Motion of Long Water Waves 



. dij /I , , n+l , n+l nfl 



ZA' 



n+l 



M-lj 



l^i-lj-l " ^i + lj*l^i + lj+l "^ ^i-lj+l ^i-lj+l 



■*'^i + lj-l^i + lj-l - ^i-lj-l^i-lj-|) - dij+|Vij*| 



+ ZdjjVjj - djj.,Vjj.|) 



2 



dii / 1 / n+l n*l n + l , n+l 



- -^\4 <^i+lj+l - ^i-lj+l - ^i+lj-l + ^i-lj-l 



- Vjj+i + Zvjj - Vij.i) 



Now, Eq, (20) must be solved for all j simultaneously for a given 1. 

 Again we use the tridiagonal-matrix equation solver. The process 

 is repeated for each i until the v"* are known. 



Fourth, the continuity Eq, (6) is used to obtain a difference 

 equation for the r\ values at the n+l level. The u and v 

 values are used in an equation that, as suggested by Peregrine [1967] , 

 uses an average for u and v values, but forward differences for 

 T| values. The result, rearranged for computation, is the explicit 

 equation 



n+l _ ^t / Uij + uij / , . „ ■, « \ 



+ 2 ^"^ij "^^ij><^i*lj +^i + lj - ^i-lj - Vlj 

 , , n+l n + l V 



+ Vij^, +Vij^, - Vjj., - Vjj.,) 



n+l 

 + ^V^<^iH +^iJ4l-dij-|-^ij-|)) (21) 



This equation is used to compute t^-'^' for 2^1^ M-1 , 2 ^ j ^ N- 1 , 

 and then the first iteration, the predictor iteration, at the n 

 time level is complete. 



If now the second through fourth steps above are repeated with 



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