A typical solution on such a domain, with the grids used, looks like:

- Start with a uniform base grid
- Compute solution on that grid
- Compute space monitor (based on the curvature)
- Where needed refine by bisection in every coordinate direction
- This gives a second grid (enclosed by the first one):
- also uniform
- cell-width halved

- Apply the same algorithm on the second grid
- This results in a series of nested grids where the PDE-system is solved in succession
- Solution at the finest-mesh points is used

- Space discretization on a 9-point stencil (because of cross-derivatives). Second-order finite differences.
- Resulting system of DAEs is solved with an implicit time-integrator (BDF2) with variable stepsizes.

J.G. Blom, R.A. Trompert and J.G. Verwer

CWI Report NM-R9403.

VLUGR3: A Vectorizable Adaptive Grid Solver for PDEs in 3D.

I. Algorithmic Aspects and Applications,

J.G. Blom and J.G. Verwer

*Appl. Numer. Math.*, Vol. 16, pp. 129-156 (1994).

CWI Report
NM-R9404.

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