Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. (arXiv:0910.1046v1 [math.NA])

For linear and fully non-linear diffusion equations of

Bellman-Isaacs 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
non-diagonal 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.

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Original source : http://arxiv.org/abs/0910.1046...