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

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

2009-10-06

arxiv.org/abs/0910.1046v2

Mathematics

Numerical Analysis

Scientific paper

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 general diffusions with coefficient matrices that may be non-diagonal dominant and degenerate. In general such schemes have to have a wide stencil. Besides providing a unifying framework for several known first order accurate schemes, our class of schemes includes more efficient versions, and new schemes that are second order accurate in space and converge 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 in the general case and robust error estimates in the convex case. The methods are extensively tested.

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