Résumé
The fractal conservation law partial derivative(t)u + partial derivative(x) (f( u)) + (-Delta)(alpha/2)u = 0 changes characteristics as alpha -> 2 from non-local and weakly diffusive to local and strongly diffusive. In this paper we present a corrected finite difference quadrature method for (-Delta)(alpha/2) with alpha epsilon [0, 2], combined with usual finite volume methods for the hyperbolic term, that automatically adjusts to this change and is uniformly convergent with respect to alpha epsilon [eta, 2] for any eta > 0. We provide numerical results which illustrate this asymptotic-preserving property as well as the non-uniformity of previous finite difference or finite volume type of methods.