A Sequence of Integro-Differential Equations Approximating a Viscous Porous Medium Equation

We consider a sequence of particular integro-differential equations, whose solutions ρN converge as N → ∞ to the solution ρ of a viscous porous medium equation. First, it is demonstrated that under suitable regularity conditions the functions ρN are smooth uniformly in N ∈ N. Furthermore, an asymptotic expansion for ρN as N → ∞ is provided, which precisely describes the convergence to ρ. The results of this paper are needed in particular for the numerical simulation of a viscous porous medium equation by a particle method.