High-order monotone finite-volume schemes for 1D elliptic problems

When solving numerically an elliptic problem, it is important in most applications that the scheme used preserves the positivity of the solution. When using finite volume schemes on deformed mesh, the question has been solved rather recently. Such schemes are usually (at most) second order convergent, and nonlinear. On the other hand, many high-order schemes have been proposed, that do not ensure positivity of the solution. In this paper we propose a very high-order monotone (that is, positivity preserving) numerical method for elliptic problems in 1D. We prove that this method converges to an arbitrary order and is indeed monotone. We also show how to handle discontinuous sources or diffusion coefficients, while keeping the order of convergence. We assess the new scheme, on several test problems, with arbitrary (regular, distorted, random) meshes.