Dirichlet-to-Neumann mappings and finite-differences for anisotropic diffusion

Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

“…-one consists in proceeding through "proxies", for instance the preservation of auxiliary quantities (systems with "involutions" in Dafermos' terminology) like vorticity or divergence constraints, see [13,[18][19][20]; -another, in starting from the expression of a multi-D (possibly local) solution, then deducing a numerical algorithm, in the spirit as e.g. [11,12] for parabolic equations, or [4]. This is what is done here, when Kirchhoff's exact solution yields a time-marching scheme for (1.1), in the form (1.2), see [27].…”

A Two-Dimensional “Flea on the Elephant” Phenomenon and its Numerical Visualization

“…1.1, analytical bounds were derived in Section 3 dealing with various perturbations of the classical quadratic harmonic potential; in particular, mass-accumulation in wider wells was justified with elementary arguments (see also [9,20,31,45]). Then, in Section 4, the 2D scheme of [16,17] is derived in a way which allows to produce error bounds for the constant coefficient case (see [52]). Such an algorithm, involving modified Bessel functions in 2D, is reminiscent of "discrete weighted means", see [15,23,25,40,52].…”

“…Such regularity in a square domain requires smooth boundary data to be supplemented by compatibility conditions (A.2) at each corner, in order to apply Theorem A.2. This is a situation closely related to "well-balanced methods" in 1D, see also [3,16,19] for 2D considerations. In particular, the modified Bessel functions contained in (4.11), which numerically allow not to split between the Laplacian and the zero-order term, moreover can fit the sharp layers appearing in the solution in case the potential becomes stiff (like "exponentialfit" methods in 1D).…”