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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Vydané v:Applied numerical mathematics Ročník 59; číslo 3; s. 707 - 722
Hlavní autori: Kulikov, G.Yu, Shindin, S.K.
Médium: Journal Article Konferenčný príspevok..
Jazyk:English
Vydavateľské údaje: Kidlington Elsevier B.V 01.03.2009
Elsevier
Predmet:
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