Rate of convergence of finite difference approximations for degenerate ordinary differential equations

Abstract

In this paper we study finite difference approximations for the following linear stationary convection-diffusion equations: \[ 1 2 σ 2 ( x ) u ( x ) + b ( x ) u ′ ( x ) − u ( x ) = − f ( x ) , x ∈ R , {1\over 2}\sigma ^2(x)u(x) + b(x)u’(x) - u(x) =-f(x),\quad x\in \mathbb {R}, \] where σ \sigma is allowed to be degenerate. We first propose a new weighted finite difference scheme, motivated by approximating the diffusion process associated with the equation in the strong sense. We show that, under certain conditions, this scheme converges with the first order rate and that such a rate is sharp. To the best of our knowledge, this is the first sharp result in the literature. Moreover, by using the connection between our scheme and the standard upwind finite difference scheme, we get the rate of convergence of the latter, which is also new.