Runge-Kutta projection methods with low dispersion and dissipation errors

被引：11

作者：

Calvo, M. ^{[1
]}

Laburta, M. P. ^{[1
]}

Montijano, J. I. ^{[1
]}

Randez, L. ^{[1
]}

机构：

[1] Univ Zaragoza, Dept Matemat Aplicada, IUMA, E-50009 Zaragoza, Spain

来源：

ADVANCES IN COMPUTATIONAL MATHEMATICS | 2015年 / 41卷 / 01期

关键词：

Initial value problems; Numerical geometric integration; Projection methods; Dispersion error; Explicit Runge-Kutta methods; GEOMETRIC INTEGRATION; OSCILLATING SOLUTIONS; RIGID-BODY; ODES; INVARIANTS; PRESERVE;

D O I：

10.1007/s10444-014-9355-2

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper new one-step methods that combine Runge-Kutta (RK) formulae with a suitable projection after the step are proposed for the numerical solution of Initial Value Problems. The aim of this projection is to preserve some first integral in the numerical integration. In contrast with standard orthogonal projection, the direction of the projection at each step is obtained from another suitable embedded formula so that the overall method is affine invariant. A study of the local errors of these projection methods is carried out, showing that by choosing proper embedded formulae the order can be increased for the harmonic oscillator. Particular embedded formulae for the third order method by Bogacki and Shampine (BS3) are provided. Some criteria to get appropriate dynamical directions for general problems as well as sufficient conditions that ensure the existence of RK methods embedded in BS3 according to them are given. Finally, some numerical experiments to test the behaviour of the new projection methods are presented.

引用

页码：231 / 251

页数：21

共 25 条

[1]

[Anonymous], 1989, APPL MATH LETT

[2] Energy-preserving methods for Poisson systems [J].

Brugnano, L. ;

Calvo, M. ;

Montijano, J. I. ;

Randez, L. .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2012, 236 (16) :3890-3904

[3] Line integral methods which preserve all invariants of conservative problems [J].

Brugnano, Luigi ;

Iavernaro, Felice .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2012, 236 (16) :3905-3919

[4] A simple framework for the derivation and analysis of effective one-step methods for ODEs [J].

Brugnano, Luigi ;

Iavernaro, Felice ;

Trigiante, Donato .

APPLIED MATHEMATICS AND COMPUTATION, 2012, 218 (17) :8475-8485

[5]

Brugnano L, 2009, AIP CONF PROC, V1168, P715, DOI 10.1063/1.3241566

[6] On the preservation of invariants by explicit Runge-Kutta methods [J].

Calvo, M. ;

Hernandez-Abreu, D. ;

Montijano, J. I. ;

Randez, L. .

SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2006, 28 (03) :868-885

[7] Projection methods preserving Lyapunov functions [J].

Calvo, M. ;

Laburta, M. P. ;

Montijano, J. I. ;

Randez, L. .

BIT NUMERICAL MATHEMATICS, 2010, 50 (02) :223-241

[8] Approximate preservation of quadratic first integrals by explicit Runge-Kutta methods [J].

Calvo, M. ;

Laburta, M. P. ;

Montijano, J. I. ;

Randez, L. .

ADVANCES IN COMPUTATIONAL MATHEMATICS, 2010, 32 (03) :255-274

[9] Explicit Runge-Kutta methods for initial value problems with oscillating solutions [J].

Calvo, M ;

Franco, JM ;

Montijano, JI ;

Randez, L .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 1996, 76 (1-2) :195-212

[10] ENERGY-PRESERVING RUNGE-KUTTA METHODS [J].

Celledoni, Elena ;

McLachlan, Robert I. ;

McLaren, David I. ;

Owren, Brynjulf ;

Quispel, G. Reinout W. ;

Wright, William M. .

ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 2009, 43 (04) :645-649

← 1 2 3 →