Browse > Article
http://dx.doi.org/10.14317/jami.2015.657

NUMERICAL ANALYSIS OF LEGENDRE-GAUSS-RADAU AND LEGENDRE-GAUSS COLLOCATION METHODS  

CHEN, DAOYONG (Department of Mathematics, Shanghai Normal University)
TIAN, HONGJIONG (Department of Mathematics, Shanghai Normal University)
Publication Information
Journal of applied mathematics & informatics / v.33, no.5_6, 2015 , pp. 657-670 More about this Journal
Abstract
In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.
Keywords
Collocation method; Runge-Kutta method; differential quadrature method; Legendre polynomial;
Citations & Related Records
연도 인용수 순위
  • Reference
1 C. Shu and Y.T. Chew, On the equivalence of generalized differential quadrature and highest order finite difference scheme, Comput. Methods Appl. Mech. Engin. 155 (1998), 249-260.   DOI
2 A. Kiliçman, I. Hashim, M. Tavassoli Kajani and M. Maleki, On the rational second kind Chebyshev pseudospectral method for the solution of the Thomas-Fermi equation over an infinite interval, J. Comput. Appl. Math. 257 (2014), 79-85.   DOI
3 J.D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, John Wiley & Sons. Chichester, UK. 1991.
4 S.P. Nørsett, C-polynomials for rational approximation to the exponential function, Numer. Math. 25 (1975), 39-56.   DOI
5 E. Tohidi and A. Kiliçman, A collocation method based on the Bernoulli operational matrix for solving nonlinear BVPs which arise from the problems in calculus of variation, Math. Prob. Eng. 2013 (2013), Article ID 757206, 9 pages.
6 M. Tavassoli Kajani, M. Maleki and A. Kiliçman, A multiple-step legendre-gauss collocation method for solving volterra's population growth model, Math. Prob. Eng. 2013 (2013), Article ID 783069, 6 pages.
7 Z. Wang and B. Guo, Legendre-Gauss-Radau collocation method for solving initial value problems of first order ordinary differential equations, J. Sci. Comput. 52 (2012), 226-255.   DOI
8 C.W. Bert, X. Wang and A.G. Striz, Differential quadrature for static and free vibration analyses of anisotropic plates, Inter. J. Solids Struct. 30 (1993), 1737-1744.   DOI
9 K. Wright, Some relationships between implicit Runge-Kutta, collocation and Lanczos τ methods, and their stability properties, BIT. 10 (1970), 217-227.   DOI
10 L. Collatz, The Numerical Treatment of Differential Equations, 2nd English Ed., Springer, Berlin, 1960.
11 J.C. Butcher, The role of orthogonal polynomials in numerical differential equations, J. Comput. Appl. Math. 43 (1992), 231-242.   DOI
12 J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons Ltd, 2008.
13 T.C. Fung, Solving initial value problems by differential quadrature method-Part 1: Erst order equations, Inter. J. Numer. Methods Engin. 50 (2001), 1411-1427.   DOI
14 T.C. Fung, Solving initial value problems by differential quadrature method-Part 2: second and higher order equations, Inter. J. Numer. Methods Engin. 50 (2001), 1429-1454.   DOI
15 T.C. Fung, On the equivalence of the time domain differential quadrature method and the dissipative Runge-Kutta collocation method, Inter. J. Numer. Methods Engin. 53 (2002), 409-431.   DOI
16 C.W. Bert and M. Malik, Differential quadrature method in computational mechanics: a review, Appl. Mech. Rev. 49 (1996), 1-28.   DOI
17 B. Guo and Z. Wang, Legendre-Gauss collocation methods for ordinary differential equations, Adv. Comput. Math. 30 (2009), 249-280.   DOI
18 E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd Edition, Springer-Verlag, Berlin 1993.
19 E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd Edition, Springer-Verlag, 1996.
20 R. Bellman and J. Casti, Differential quadrature and long-term integration, J. Math. Anal. Appl. 34 (1971), 235-238.   DOI
21 R. Barrio, On the A-stability of Runge-Kutta collocation methods based on orthogonal polynomials, SIAM J. Numer. Anal. 36 (1999), 1291-1303.   DOI