TY - JOUR

T1 - Time-point relaxation Runge-Kutta methods for ordinary differential equations

AU - Bellen, A.

AU - Jackiewicz, Zdzislaw

AU - Zennaro, M.

N1 - Funding Information: Italy. * These authors’ research was supported by the CNR Progetto Finalizzato “Sistemi Informatici Parallelo” and MURST. ** This author’s research was supported by the CNR and the NSF under grant DMS-8900411.

PY - 1993/4/8

Y1 - 1993/4/8

N2 - We investigate convergence, order, and stability properties of time-point relaxation Runge-Kutta methods for systems of ordinary differential equations. These methods can be implemented in Gauss-Jacobi, or Gauss-Seidel modes denoted by TRGJRK(k) or TRGSRK(k), respectively, where k stands for the number of Picard-Lindelöf iterations. As k → ∞, these modes tend to the same one-step method for ordinary differential equations called diagonal split Runge-Kutta (DSRK) method. It is proved that these methods are convergent with order min{k, p}, where p is the order of the underlying Runge-Kutta method. Recurrence relations resulting from application of TRGJRK(k), TRGSRK(k) and DSRK methods to the two-dimensional test system u′ = λu - μν, ν′ = μu + λν, t ≥ 0, where λ and μ are real parameters, are derived and stability regions in the (λ, μ)-plane are plotted for some methods using a variant of the boundary locus method. In most cases stability regions increase as the number of Picard-Lindelöf iterations k becomes larger.

AB - We investigate convergence, order, and stability properties of time-point relaxation Runge-Kutta methods for systems of ordinary differential equations. These methods can be implemented in Gauss-Jacobi, or Gauss-Seidel modes denoted by TRGJRK(k) or TRGSRK(k), respectively, where k stands for the number of Picard-Lindelöf iterations. As k → ∞, these modes tend to the same one-step method for ordinary differential equations called diagonal split Runge-Kutta (DSRK) method. It is proved that these methods are convergent with order min{k, p}, where p is the order of the underlying Runge-Kutta method. Recurrence relations resulting from application of TRGJRK(k), TRGSRK(k) and DSRK methods to the two-dimensional test system u′ = λu - μν, ν′ = μu + λν, t ≥ 0, where λ and μ are real parameters, are derived and stability regions in the (λ, μ)-plane are plotted for some methods using a variant of the boundary locus method. In most cases stability regions increase as the number of Picard-Lindelöf iterations k becomes larger.

KW - Ordinary differential equations

KW - Runge-Kutta method

KW - stability analysis

KW - time-point relaxation

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U2 - 10.1016/0377-0427(93)90269-H

DO - 10.1016/0377-0427(93)90269-H

M3 - Article

SN - 0377-0427

VL - 45

SP - 121

EP - 137

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

IS - 1-2

ER -