Minimal Stencils for Monotony or Causality Preserving Discretizations of Anisotropic PDEs
Résumé
We consider discretizations of anisotropic diffusion and of the anisotropic eikonal equa-tion, on two dimensional cartesian grids, which preserve their structural properties: mono-tonicity of diffusion, causality of the eikonal equation. These two PDEs embed geometric information, in the form of a field of diffusion tensors and of a Riemannian metric respectively. Common knowledge is that, when these tensors are strongly anisotropic, monotonous or causal discretizations of these PDEs cannot be strictly local: numerical schemes need to involve interactions between each point and the elements of a stencil, which is not limited to its immediate neighbors on the discretization grid. Using tools from discrete geometry we identify the smallest valid stencils, in the sense of convex hull inclusion. We also estimate, for a fixed condition number but a random orientation of the tensors, the worst case and the average case radius of these minimal stencils, which is relevant for numerical error analysis.
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