Semi-Lagrangian schemes for linear and fully non-linear diffusion equations

For linear and fully non-linear diffusion equations of BellmanIsaacs type, we introduce a class of monotone approximation schemes relying on monotone interpolation. As opposed to classical numerical methods, these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices. Such schemes have to have a wide stencil in general. Besides providing a unifying framework for several known first order accurate schemes, our class of schemes also includes more efficient versions, and a new second order scheme that converges only for essentially monotone solutions. The methods are easy to implement and analyze, and they are more efficient than some other known schemes. We prove stability and convergence of the schemes in the general case, and provide error estimates in the convex case which are robust in the sense that they apply to degenerate equations and non-smooth solutions. The methods are extensively tested.