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Mathematics of Computation

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Computation of optimal monotonicity preserving general linear methods

Author: David I. Ketcheson
Journal: Math. Comp. 78 (2009), 1497-1513
MSC (2000): Primary 65L06
Published electronically: January 22, 2009
MathSciNet review: 2501060
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Abstract: Monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution. We formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations, and propose an efficient algorithm for its solution. This algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and/or stages. The optimality of some recently proposed methods is verified, and many more efficient methods are found. We use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type, including methods for hyperbolic PDEs that use downwind-biased operators.

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Additional Information

David I. Ketcheson
Affiliation: Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-2420

Received by editor(s): May 13, 2008
Received by editor(s) in revised form: August 19, 2008
Published electronically: January 22, 2009
Additional Notes: This work was supported by a U.S. Department of Energy Computational Science Graduate Fellowship under grant number DE-FG02-97ER25308, and by AFOSR grant number FA9550-06-1-0255.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.