Numerical approximations to interface curves for a porous media equation

Standing pulse-like solutions of a spatially aggregating population model

“…Because of the finite propagation property, there appear interface curves that divide the half plane R x (0, ~) into two regions {(x, t); u(x, t) > 0} and {(x, t); u(x, t) = 0} (for instance, Aronson [5], Caffarelli and Friedman [10] and Knerr [25]). Graveleau and Jamet [15] and Tomoeda and Mimura [43] have proposed finite difference schemes where the degeneracy of diffusion is taken into account. Schemes proposed by DiBenedetto and Hoff [12], Mimura et al [28] and Hoff [20] approximate the interface curves as well as the value of the unknown function.…”

A numerical approach to interface curves for some nonlinear diffusion equations

“…From a numerical point of view, there are several finite difference schemes to (1.5), (1.6) with F(v)=0, that is, the porous media equation, proposed by Baklanovskaya [2], Graveleau and Jamet [11], DiBenedetto and Hoff [10], Tomoeda and Mimura [25] and Hoff [13]. Among these, we should mention two schemes which approximate both the solution and the interfaces.…”

Finite difference method for a fractional porous medium equation

“…In the limit σ → 2 we recover the standard Porous Medium Equation ∂u ∂t + ∆(|u| m−1 u) = 0, for which the numerical solution has been studied by many authors, either in itself of as part of the study of the class degenerate parabolic equations, see e.g. [1,3,11,13,15,16,18,20,21,24] for the earlier literature.…”