Adaptive nested implicit Runge–Kutta formulas of Gauss type

This paper deals with a special family of implicit Runge–Kutta formulas of orders 2, 4 and 6. These methods are of Gauss type; i.e., they are based on Gauss quadrature formulas of orders 2, 4 and 6, respectively. However, the methods under discussion have only explicit internal stages that lead to c...

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Vydáno v:	Applied numerical mathematics Ročník 59; číslo 3; s. 707 - 722
Hlavní autoři:	Kulikov, G.Yu, Shindin, S.K.
Médium:	Journal Article Konferenční příspěvek
Jazyk:	angličtina
Vydáno:	Kidlington Elsevier B.V 01.03.2009 Elsevier
Témata:	Almost symplectic integration Approximations and expansions Exact sciences and technology Gauss-type methods Local error estimation Mathematical analysis Mathematics Measure and integration Nested implicit Runge–Kutta formulas Numerical analysis Numerical analysis. Scientific computation Numerical approximation Ordinary differential equations Real functions Sciences and techniques of general use 65L05 65L06 Nested implicit Runge–Kutta formulas Gauss-type methods Ordinary differential equations Local error estimation Almost symplectic integration Numerical integration Error estimation Differential equation Gauss method Runge Kutta method Gauss formula Nested implicit Runge-Kutta formulas Implementation Numerical approximation Numerical analysis Cubature Applied mathematics Quadrature formula Hamiltonian 65L06 Ordinary differential equations
ISSN:	0168-9274, 1873-5460
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Popis
Shrnutí:	This paper deals with a special family of implicit Runge–Kutta formulas of orders 2, 4 and 6. These methods are of Gauss type; i.e., they are based on Gauss quadrature formulas of orders 2, 4 and 6, respectively. However, the methods under discussion have only explicit internal stages that lead to cheap practical implementation. Some of the stage values calculated in a step of the numerical integration are of sufficiently high accuracy that allows for dense output of the same order as the Runge–Kutta formula used. On the other hand, the methods developed are A-stable, stiffly accurate and symmetric. Moreover, they are conjugate to a symplectic method up to order 6 at least. All of these make the new methods attractive for solving nonstiff and stiff ordinary differential equations, including Hamiltonian and reversible problems. For adaptivity, different strategies of error estimation are discussed and examined numerically.
ISSN:	0168-9274 1873-5460
DOI:	10.1016/j.apnum.2008.03.019