NDLI: Explicit Runge–Kutta pairs with lower stage-order

Content Provider	Springer Nature Link
Author	Verner, J. H.
Copyright Year	2013
Abstract	Explicit Runge–Kutta pairs of methods of successive orders of accuracy provide effective algorithms for approximating solutions to nonstiff initial value problems. For each explicit RK method of order of accuracy p, there is a minimum number s $_{ p }$ of derivative evaluations required for each step propagating the numerical solution. For p ≤ 8, Butcher has established exact values of s $_{ p }$, and for p > 8, his work establishes lower bounds; otherwise, upper bounds are established by various published methods. Recently, Khashin has derived some new methods numerically, and shown that the known upper bound on s $_{9}$ for methods of order p = 9 can be reduced from 15 to 13. His results motivate this attempt to identify parametrically exact representations for coefficients of such methods. New pairs of methods of orders 5,6 and 6,7 are characterized in terms of several arbitrary parameters. This approach, modified from an earlier one, increases the known spectrum of types of RK pairs and their derivations, may lead to the derivation of new RK pairs of higher-order, and possibly to other types of explicit algorithms within the class of general linear methods.
Ending Page	577
Page Count	23
Starting Page	555
File Format	PDF
ISSN	10171398
e-ISSN	15729265
Journal	Numerical Algorithms
Issue Number	3
Volume Number	65
Language	English
Publisher	Springer US
Publisher Date	2013-11-16
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Stability and convergence of numerical methods Algorithms Algebra Multistep, Runge-Kutta and extrapolation methods Numerical Analysis Numeric Computing Runge–Kutta pairs Order conditions Stage-order Initial value problems Theory of Computation Local error estimation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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