On numerical methods and error estimates for degenerate fractional convection–diffusion equations

Abstract

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods -even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations.

A Uniformly Converging Scheme for Fractal Conservation Laws

“…The last term in (7) contains the classical discretization of ϕ ′′ (x i ) and is the new correction term compared with the discretisations of [13,10,8]. This discretisations fit with the generic framework of [13] from which we can conclude:…”

“…Except for the correction term, the scheme is similar to the schemes of [13,8] and of [10] with P 0 -elements. It is monotone, conservative, and converges in L 1…”

“…The first and simplest idea to obtain a monotone discretization of L α for α ∈ (0, 2) is to discretize the integral in (3) using a simple (weighted) midpoint type quadrature rule, see e.g. [13,10,8]…”

“…Here and elsewhere, the convergence (δ x, δt) → 0 is always taken under a standard CFL condition depending on the definition of the convective flux F in (5) (see e.g. [13,10,8]). That this formulation of the asymptotic-preserving property is very general and does not require an explicit error estimate independent on α.…”

“…Probabilistic methods have been studied in [21,24], but must be applied to the equation satisfied by ∂ x u α in order to avoid noisy results, and recovering from this a numerical approximation of u α may be challenging in dimension greater than 1. Deterministic methods for (1) like finite difference, volume, and element methods (discontinuous Galerkin) are given in [13,8,10], while a high order spectral vanishing viscosity method is introduced in [9]. The latter method and its analysis is very different from the former three methods, with convergence and (non-optimal) error estimates that are independent of α ∈ (0, 2).…”