A conditionally \(P\)-stable fourth-order exponential-fitting method for \(y''=f(x,y)\). (English) Zbl 0934.65079

Ixaru, L. Gr.; Paternoster, B.

J. Comput. Appl. Math. 106, No. 1, 87-98 (1999).

The authors define the exponential-fitting version of the fourth-order two-step method introduced by M. M. Chawla [BIT 21, 190-193 (1981; Zbl 0457.65053)]. By introducing the concept of ‘ ‘ conditional \(P\)-stability’ ’ of a family of exponential-fitting methods and the parameter \( \theta_{max} \) associated to a conditioned \(P\)-stable family they prove that the new version of Chawla algorithm is conditionally \(P\)-stable with \( \theta_{\max} = 3.4 \). Moreover, a numerical test is reported in order to make out the following two points: the new method with \( \theta < 3.4 \) can be effectively applied to solving stiff problems; due to the exponential-fitting feature, on problems with oscillating solution the method produces more accurate results than its polynomial-fitting companion.

Reviewer: R.Fazio(Messina)

MSC:

65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05	Numerical methods for initial value problems involving ordinary differential equations
65L20	Stability and convergence of numerical methods for ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems

Keywords:

hybrid multistep method; exponential fitting; \(P\)-stability; algorithm; stiff problems; oscillating solution

Citations:

Zbl 0457.65053

Software:

EXPFIT4

References:

[1]	Chawla, M. M., Two-step fourth-order P-stable methods for second order differential equations, BIT, 21, 190-193 (1981) · Zbl 0457.65053
[2]	Coleman, J. P., Numerical methods for \(y” = f(x,y)\), via rational approximations to the cosine, IMA J. Numer. Anal., 9, 145-165 (1989) · Zbl 0675.65072
[3]	Coleman, J. P.; Ixaru, L. Gr., P-stability and exponential-fitting methods for \(y”=f(x,y)\), IMA J. Numer. Anal., 16, 179-199 (1996) · Zbl 0847.65052
[6]	Ixaru, L. Gr.; De Meyer, H.; Vanden Berghe, G.; Van Daele, M., EXPFIT4 - a FORTRAN program for the numerical solution of nonlinear second-order initial-value problems, Comput. Phys. Commun., 100, 71-80 (1997) · Zbl 0927.65099
[7]	Lambert, J. D.; Watson, I. A., Symmetric multistep methods for periodic initial value problems, J. Inst. Math. Appl., 18, 189-202 (1976) · Zbl 0359.65060
[8]	Raptis, A. D.; Simos, T. E., A four-step phase-fitted method for the numerical integration of second-order initial-value problems, BIT, 31, 160-168 (1991) · Zbl 0726.65089
[9]	Simos, T. E.; Dimas, E.; Sideridis, A. B., A Runge-Kutta-Nyström method for the numerical integration of special second-order periodic initial-value problems, J. Comput. Appl. Math., 51, 317-326 (1994) · Zbl 0872.65066
[10]	Simos, T. E.; Raptis, A. D., Numerov-type methods with maximal phase-lag for the numerical integration of the one-dimensional Schrödinger equation, Computing, 45, 175-181 (1990) · Zbl 0721.65045