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

Authors: Kristian Debrabant and Espen R. Jakobsen
Journal: Math. Comp. 82 (2013), 1433-1462
MSC (2010): Primary 65M12, 65M15, 65M06, 35K10, 35K55, 35K65, 49L25, 49L20
Published electronically: December 20, 2012
MathSciNet review: 3042570
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Abstract: For linear and fully non-linear diffusion equations of Bellman-Isaacs type, we introduce a class of approximation schemes based on differencing and interpolation. As opposed to classical numerical methods, these schemes work for general diffusions with coefficient matrices that may be non-diagonal dominant and arbitrarily 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 new first and higher order versions. The methods are easy to implement and more efficient than some other known schemes. We prove consistency and stability of the methods, and for the monotone first order methods, we prove convergence in the general case and robust error estimates in the convex case. The methods are extensively tested.

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Additional Information

Kristian Debrabant
Affiliation: University of Southern Denmark, Department of Mathematics and Computer Science, Campusvej 55, 5230 Odense M, Denmark

Espen R. Jakobsen
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway

Keywords: Monotone approximation schemes, difference-interpolation methods, stability, convergence, error bound, degenerate parabolic equations, Hamilton-Jacobi-Bellman equations, viscosity solution.
Received by editor(s): October 20, 2009
Received by editor(s) in revised form: September 6, 2011, and October 15, 2012
Published electronically: December 20, 2012
Additional Notes: The second author was supported by the Research Council of Norway through the project “Integro-PDEs: Numerical methods, Analysis, and Applications to Finance”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.