SPECTRAL DEFERRED CORRECTION METHODS FOR SECOND-ORDER PROBLEMS

被引:0
作者
Akramov, Ikrom [1 ]
Gotschel, Sebastian [1 ]
Minion, Michael [2 ]
Ruprecht, Daniel [1 ]
Speck, Robert [3 ]
机构
[1] Tech Univ Hamburg, Lehrstuhl Computat Math, Hamburg, Germany
[2] Lawrence Berkeley Natl Lab, Berkeley, CA 94720 USA
[3] Forschungszentrum Julich, Julich Supercomp Ctr, Julich, Germany
关键词
spectral deferred corrections (SDC); Picard iteration; collocation method; velocity-Verlet; stability; convergence; KUTTA-NYSTROM METHODS; RUNGE-KUTTA; ORDER; CONVERGENCE; EXPLICIT;
D O I
10.1137/23M1592596
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit, or implicit-explicit Euler and other low- order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second- order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocityVerlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta--Nystro"\m method.
引用
收藏
页码:A1690 / A1713
页数:24
相关论文
共 37 条
[1]  
AGARWAL P., 2018, Fixed Point Theory in Metric Spaces: Recent Advances and Applications, DOI DOI 10.1007/978-981-13-2913-5
[2]   Modified Chebyshev-Picard Iteration Methods for Orbit Propagation [J].
Bai, Xiaoli ;
Junkins, John L. .
JOURNAL OF THE ASTRONAUTICAL SCIENCES, 2011, 58 (04) :583-613
[3]   Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nystrom methods [J].
Blanes, S ;
Moan, PC .
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2002, 142 (02) :313-330
[4]  
Boris J. P., 1972, 5th European Conference on Controlled Fusion and Plasma Physics. Vol.II
[5]   A multirate time integrator for regularized Stokeslets [J].
Bouzarth, Elizabeth L. ;
Minion, Michael L. .
JOURNAL OF COMPUTATIONAL PHYSICS, 2010, 229 (11) :4208-4224
[6]   GEONIUM THEORY - PHYSICS OF A SINGLE ELECTRON OR ION IN A PENNING TRAP [J].
BROWN, LS ;
GABRIELSE, G .
REVIEWS OF MODERN PHYSICS, 1986, 58 (01) :233-311
[7]  
Caklovic G, 2022, Arxiv, DOI [arXiv:2203.04086, 10.48550/arXiv.2203.04086, DOI 10.48550/ARXIV.2203.04086]
[8]   ON THE CONVERGENCE OF SPECTRAL DEFERRED CORRECTION METHODS [J].
Causley, Mathew F. ;
Seal, David C. .
COMMUNICATIONS IN APPLIED MATHEMATICS AND COMPUTATIONAL SCIENCE, 2019, 14 (01) :33-64
[9]  
Chicone C., 2006, Texts Appl. Math., V34, DOI DOI 10.1007/0-387-35794-7
[10]   INTEGRAL DEFERRED CORRECTION METHODS CONSTRUCTED WITH HIGH ORDER RUNGE-KUTTA INTEGRATORS [J].
Christlieb, Andrew ;
Ong, Benjamin ;
Qiu, Jing-Mei .
MATHEMATICS OF COMPUTATION, 2010, 79 (270) :761-783