Computation of optimal monotonicity preserving general linear methods

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...

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Bibliographic Details
Published in:Mathematics of computation 2009-07, Vol.78 (267), p.1497-1513
Main Author: Ketcheson, David I.
Format: Article
Language:English
Subjects:
Algorithms
Cauchy problem
Coefficients
Exact sciences and technology
Integers
Linear programming
Mathematical analysis
Mathematical monotonicity
Mathematics
Methods of scientific computing (including symbolic computation, algebraic computation)
Numbers
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods
Ordinary differential equations
Partial differential equations
Polynomials
Runge Kutta method
Sciences and techniques of general use
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Summary: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. \par
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-09-02209-1