Elliptic Methods



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Elliptic Methods

 

Elliptic methods are based on the solution of the boundary value problem [Hil88a] , . In the computational domain, the corresponding transformed equations (3.3-18) - (3.3-19) have to be solved.

 

 

 

This system of nonlinear elliptic PDEs is discretized on the geometric grid using nine-point finite differences. The resulting system of coupled nonlinear algebraic equations is solved with a nonlinear SOR Algorithm [Ort70].

A superimposed iteration scheme (3.3-21) - (3.3-23) determines the source functions and at the boundaries which are used to avoid clustering of gridlines at concave corners, and to allow orthogonality control at the boundaries [Hil88a], [Bau90].

 

 

 

Here denotes the angle of intersection of gridlines (3.3-24) (which should be equal to the required value ), and resembles the grid spacing (3.3-25).

 

 

The upper signs in (3.3-22) and (3.3-23) relate to the north boundary, the lower signs to the south boundary. For the east and west boundaries, the correction terms for and are exchanged. In the interior, and are obtained by transfinite interpolation. In his original work [Hil88a] Hilgenstock proposed the - and -values at the boundary should decay exponentially into the interior of the domain. This turned out inferior to the transfinite interpolation.



Martin Stiftinger
Wed Oct 19 13:03:34 MET 1994