MINIMAL STENCILS FOR DISCRETIZATIONS OF ANISOTROPIC PDEs PRESERVING CAUSALITY OR THE MAXIMUM PRINCIPLE

摘要

We consider discretizations of anisotropic diffusion and of the anisotropic eikonal equation, on two-dimensional cartesian grids, which preserve their structural properties: the maximum principle for diffusion, and causality for the eikonal equation. These two PDEs embed geometric information in the form of a field of diffusion tensors and of a Riemannian metric, respectively. It is common knowledge 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 tensor orientation, the worst case and average case radius of these minimal stencils, which is relevant for numerical error analysis.