Abstract
Two 2-step P-stable methods for the numerical solution of special second order initial value problems are developed in this paper. One is of the Numerov type and of algebraic order 4 and the other is of the Runge-Kutta type and of algebraic order 6. Each of these methods has free parameters which may be chosen so that they are P-stable and have phase-lag of order infinity. The methods are used on problems with oscillatory solutions. The results indicate that these techniques are more efficient than other well known methods.
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Landau, L.D., Lifshitz, F.M.: Quantum Mechanics. Pergamon, New York, 1965
Liboff, R.L.: Introductory quantum mechanics. Addison-Wesley Publishing Company, 1980
Raptis, A.D., Allison, A.C.: Exponential-fitting methods for the numerical solution of the Schrödinger equation. Comput. Phys. Commun. 14 (1978) 1–5
Cash, J.R., Raptis, A.D., Simos, T.E.: A sixth-order exponentially fitted method for the numerical solution of the radial Schrödinger equation. J. Comput. Phys. 91 (1990) 413–423
Simos, T.E.: A two-step method with phase-lag of order infinity for the numerical integration of second order periodic initial value problem. Inter. J. Comput. Math. 39 (1991) 135–140
Simos, T.E.: Two-step almost P-stable complete in phase methods for the numerical integration of second order periodic initial value problems. Inter. J. Comput. Math. 46 (1992) 77–85
Lambert, J.D., Watson, I.A.: Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Applic. 18 (1976) 189–202
Cash, J.R.: High order P-stable formulae for the numerical integration of periodic initial value problems. Numer. Math. 37 (1981) 355–370
Chawla, M.M., Rao, P.S.: High accuracy methods for y″=f(x,y). IMA J. Numer. Anal. 5 (1985) 215–220
Hairer, E.: Uncoditionally stable methods for second order differential equations. Numer. Math. 32 (1979) 373–379
Coleman, J.P.: Numerical methods for y″=f(x, y) via rational approximation for the cosine. IMA J. Numer. Anal. 9 (1989) 145–165
Brusa, L., Nigro, L.: A one-step method for direct integration of structural dynamical equations. Int. J. Numer. Methods Engrg. 15 (1980) 685–699
Thomas, R.M.: Phase properties of high almost P-stable formulae. BIT 24 (1984) 225–238
Simos, T.E., Mousadis, G.: A two-step method for the numerical solution of the radial Schrödinger equation. Comput. Math. Appl. 29 (1995) 31–37
Coleman, J.P., Ixaru, L.Gr.: P-stability and exponential-fitting methods for y″=f(x,y). University of Durham, Numerical Analysis Report NA-94/05, Durham 1994
Kramarz, L.: Stability of colocation methods for the numerical solution of y″=f(x,y). BIT 20 (1980) 215–222
Simos, T.E.: New variable-step procedure for the numerical integration of the one-dimensional Schrödinger equation. J. Comput. Phys. 108 (1993) 175–179
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© 1997 Springer-Verlag Berlin Heidelberg
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Williams, P.S., Simos, T.E. (1997). Two-step P-stable methods with phase-lag of order infinity for the numerical solution of special second order initial value problems. In: Vulkov, L., Waśniewski, J., Yalamov, P. (eds) Numerical Analysis and Its Applications. WNAA 1996. Lecture Notes in Computer Science, vol 1196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62598-4_138
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DOI: https://doi.org/10.1007/3-540-62598-4_138
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