LAPLACE SOLVER 



To solve the Laplace equation for (|), a finite difference method based 



on boundary fitted coordinates, due to Thompson et al . [15], Is chosen. The 

 finite-difference method Involves mapping the time-dependent fluid region onto 



a fixed computational region. The coordinates E, and ri in the computational 



region are such that they obey the Laplace equation with x and y as dependent 

 variables: 



?xx + 5yy =0 (26a) 



^xx + Tiyy = (26b) 



The boundary conditions for 5 and n along a given boundary are Dirlchlet 

 if a particular mesh distribution along the boundary is prescribed. The 

 boundary condition of one coordinate is Dirlchlet and that of the other 

 coordinate is Neumann if mesh orthogonality near a particular boundary is 

 desired. 



Since all computations are to be done in the fixed (^ ,n)-computational 

 region, it is convenient to interchange the independent and dependent 

 variables. When this is done, the Equations (26a, b) for 5 and n become 



ax - 2Bx + yx =0 (27a) 



CC ?n nn 



where 



ay - 2By + Ty =0 (27b) 



CC Cn nn 



a = x2 + y2 (27c) 



n n 



= x x + y y (27d) 



y = x2 + y2 (27e) 



5 C 



16 



