VLUGR2: adaptive grid solver for PDEs in 2D

Developed in the LUGR project, which until 1994 was part of CWI's research program Mathematics & the Environment.

Problem class:

Systems of (time-dependent) partial differential equations in two space-dimensions having solutions with steep gradients in space and time. The domain can be any area that can be described by right-angled polygons.
A typical solution on such a domain, with the grids used, looks like:

Algorithms:

Local Uniform Grid Refinement:

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

Solution of PDE-system on one grid is done with the Method of Lines principle

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.

References:

Algorithm 758: VLUGR2: A Vectorizable Adaptive Grid Solver for PDEs in 2D,
J.G. Blom, R.A. Trompert and J.G. Verwer
ACM Trans. Math. Softw., Vol. 22, No. 3, pp. 302-328 (1996).
Download software from ACM TOMS mirrorsites in: Tennessee, New Jersey, Norway, England, Australia,
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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