Generalized Chebyshev Collocation Method

In this paper, we introduce a generalized Chebyshev
collocation method (GCCM) based on the generalized Chebyshev
polynomials for solving stiff systems. For employing a technique
of the embedded Runge-Kutta method used in explicit schemes, the
property of the generalized Chebyshev polynomials is used, in which
the nodes for the higher degree polynomial are overlapped with those
for the lower degree polynomial. The constructed algorithm controls
both the error and the time step size simultaneously and further
the errors at each integration step are embedded in the algorithm
itself, which provides the efficiency of the computational cost. For
the assessment of the effectiveness, numerical results obtained by the
proposed method and the Radau IIA are presented and compared.

• Junghan Kim
• Wonkyu Chung
• Sunyoung Bu
• Philsu Kim

• Generalized Chebyshev Collocation method
• Generalized Chebyshev Polynomial
• Initial value problem.

[1] E. HAIRER, G.WANNER, Solving Ordinary Differential Equations,II Stiff
and Differential-Algebraic Problems, Springer Series in Computational
Mathematics, Springer, 1996.
[2] T. HASEGAWA, T. TORII, I. NINOMIYA, Generalized Chebyshev
interpolation and its application to automatic quadrature, Math. Comp.
41(1983) pp. 537–543.
[3] T. HASEGAWA, T. TORII, H. SUGIURA, An Algorithm based on The FFT
for a Generalized Chebyshev Interpolation, Mathematics of computation,
Vol.54, Number 189(1990) pp. 195–210.
[4] L.G. IXARU, Exponentially fitted variable two-step BDF algorithm for
first order ODEs, Comput. Appl. Math. 111(1999) pp. 93–111.
[5] P. S. KIM, X. PIAO, S. D. KIM, An Error Corrected Euler Method for
Solving Stiff Problems based on Chebyshev Collocation, SIAMJ. Numer.
Anal., 48(2011) pp. 1759–1780.
[6] S. D. KIM, X. PIAO, D. H. KIM, P. S. KIM, Convergence on Error
Correction Methods for Solving Initial Value Problems, J. Comp. and
Applied Math., 236(2012) pp. 4448–4461.
[7] P. KIM, B.I. YUN, On the convergence of interpolatory-type quadrature
rules for evaluating Cauchy integrals, J. Comp. and Applied Math.,
149(2002) pp. 381–395.
[8] P. KIM, A Trigonometric Quadrature Rule for Cauchy Integrals with
Jacobi Weight, J. Approx. 108(2001) pp. 18–35.
[9] S. KIM, J. KWON, X. PIAO, P. KIM, A Chebyshev collocation method for
stiff initial value problems and its stability Kyungpook math. J. 51(2011)
pp. 435–456.
[10] A. PROTHERO, A. ROBINSON,On the stability and accuracy of one-step
methods for solving stiff systems of ordinary differential equations, Math.
Comp., 28(1974) pp. 145–162.
[11] H. RAMOS, J. VIGO-AGUIAR, A Fourth-Order Runge-Kutta Method
based on BDF-type Chebyshev Approximations, J. Comp. and Applied
Math., 204(2007) pp. 124–136.