In this talk we will introduce and analyse a class of robust numerical methods for nonlocal possibly nonlinear diffusion and convection-diffusion equations. Diffusion and convection-diffusion models are popular in Physics, Chemistry, Engineering, and Economics, and in many models the diffusion is anomalous or nonlocal. This means that the underlying “particle" distributions are not Gaussian, but rather follow more general Levy distributions, distributions that need not have second moments and can satisfy (generalised) central limit theorems. We will focus on models with nonlinear possibly degenerate diffusions like fractional Porous Medium Equations, Fast Diffusion Equations, and Stefan (phase transition) Problems, with or without convection. The solutions of these problems can be very irregular and even possess shock discontinuities. The combination of nonlinear problems and irregular solutions makes these problems challenging to solve numerically.
The methods we will discuss are monotone finite difference quadrature methods that are robust in the sense that they “always” converge. By that we mean that under very weak assumptions, they converge to the correct generalised possibly discontinuous generalised solution. In some cases we can also obtain error estimates. The plan of the talk is: 1. to give a short introduction to the models, 2. explain the numerical methods, 3. give results and elements of the analysis for pure diffusion equations, and 4. give results and ideas of the analysis for convection-diffusion equations.