A nonlinear explicit two-step fourth algebraic order method of order infinity for linear periodic initial value problems

Computer Physics Communications

Volumn 181, Issue 8, 2010, Pages 1362-1368

  Stavroyiannis, S ^a   Simos, T E ^b  

^a TECHNOLOGICAL EDUCATIONAL INSTITUTE OF KALAMATA   (Greece)

^b UNIVERSITY OF PELOPONNESE   (Greece)

Author keywords

Infinite phase lag order; P stability; Second order linear initial value problem; Vector product and quotient

Indexed keywords

ALGEBRAIC ORDER; ANALYTIC FUNCTIONS; FREQUENCY REGIMES; MULTIDIMENSIONAL PROBLEMS; NUMERICAL RESULTS; P-STABILITY; PERIODIC INITIAL VALUE PROBLEMS; PHASE LAGS; SECOND ORDERS; SECOND-ORDER LINEAR INITIAL VALUE PROBLEM; TWO STEP METHOD;

INITIAL VALUE PROBLEMS; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS; ORDINARY DIFFERENTIAL EQUATIONS; SENSITIVITY ANALYSIS; VECTORS;

ALGEBRA;

EID: 77953323491     PISSN: 00104655     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cpc.2010.04.002     Document Type: Article

References (40)

1. Lambert J.D., and Watson I.A. Symmetric multistep methods for periodic initial value problems. J. Inst. Math. Appl. 18 (1976) 189-202
2. Brusa L., and Nigro L. A one-step method for direct integration of structural dynamic equations. Internat. J. Numer. Methods Engng. 15 (1980) 685-699
3. ″ = f (x, y) via rational approximations for the cosine. IMA J. Numer. Anal. 9 (1989) 145-165
4. Ananthakrishnaiah U. P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems. Math. Comp. 49 (1987) 553-559
5. ″ = f (t, y). IMA J. Numer. Anal. 5 (1985) 215-220
6. ″ = f (t, y). J. Comput. Appl. Math. 16 (1986) 233-236
7. Hairer E. Unconditionally stable method for second order differential equations. Numer. Math. 32 (1979) 373-379
8. ″ = f (x, y). In: Mitsui T., and Shinohara Y. (Eds). Numerical Analysis of Ordinary Differential Equations and Its Applications (1995), World Scientific Publishing Co., Singapore 29-41
9. Simos T.E. A four-step method for the numerical solution of the Schrödinger equation. J. Comput. Appl. Math. 30 (1990) 251-255
10. Simos T.E., and Tsitouras Ch. A P-stable eighth-order method for the numerical integration of periodic initial value problems. J. Comput. Phys. 130 (1997) 123-128
11. Van Daele M., et al. On a class of P-stable mono-implicit methods. Appl. Numer. Math. 27 (1998) 69-82
12. ″ = f (t, y). J. Comput. Math. 13 (1995) 232-242
13. Wu X.Y. A sixth-order A-stable explicit one-step method for stiff systems. Comput. Math. Appl. 35 (1998) 59-64
14. ′ = λ y. Comput. Math. Appl. 39 5-6 (2000) 249-257
15. Wu X.Y., and Xia J.L. The vector form of a sixth-order A-stable explicit one-step method for stiff problems. Comput. Math. Appl. 39 3-4 (2000) 247-257
16. Li Q.H., and Wu X.Y. A two-step explicit P-stable method for solving second order initial value problems. Appl. Math. Comput. 138 (2003) 435-442
17. Li Q.H., and Wu X.Y. A two-step P-stable method of high phase-lag order for second order IVPs. Appl. Math. Comput. 151 (2004) 17-26
18. Li Q.H., and Wu X.Y. A two-step explicit P-stable method of high-phase lag order for linear periodic IVPs. J. Comput. Appl. Math. 200 (2007) 287-296
19. Wu X.Y., and Xia J.L. New vector forms of elemental functions with Taylor series. Appl. Math. Comput. 141 (2003) 307-312
20. Raptis A.D., and Simos T.E. A four-step phase-fitted method for the numerical integration of second order initial-value problems. BIT 31 (1991) 160-168
21. Simos T.E. A two step method with phase-lag of order infinity for the numerical integration of second order periodic initial-value problem. Int. J. Comput. Math. 39 (1991) 135-140
22. Simos T.E. An explicit almost P-stable two-step method with phase-lag of order infinity for numerical integration of second order periodic initial value problems. Appl. Math. Comput. 49 (1992) 261-268
23. Simos T.E. A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial-value problems with oscillatory solutions. Comput. Math. Appl. 25 (1993) 95-101
24. Simos T.E., and Vigo Aguiar J. A modified Runge-Kutta method with phase-lag of order infinity for the numerical solutions of the Schrödinger equation and related problems. Comp. & Chem. 25 (2001) 275-281
25. Bratsos A.G., Famelis I.Th., Kollias A., and Tsitouras Ch. Phase-fitted Numerov type methods. Appl. Math. Comput. 184 (2007) 23-29
26. van der Vyver H. A phase-fitted and amplification-fitted explicit two-step hybrid method for second order periodic initial value problems. Internat. J. Modern Phys. C 17 (2006) 663-675
27. ″ = f (x, y). J. Comput. Appl. Math. 209 (2007) 33-53
28. Anastassi Z.A., and Simos T.E. Special optimized Runge-Kutta methods for IVP's with oscillatory solutions. Internat. J. Modern Phys. C 15 (2004) 1-15
29. Anastassi Z.A., and Simos T.E. A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for numerical solution of orbital problems. New Astronom. 10 (2004) 31-37
30. ″ = f (x, y). Appl. Numer. Math. 53 (2005) 201-212
31. ″ = f (x, y). J. Comput. Appl. Math. 126 (2000) 47-75
32. van der Vyver H. Frequency evaluation for exponentially-fitted Runge-Kutta methods. J. Comput. Appl. Math. 184 (2005) 442-463
33. van der Vyver H. A fourth-order symplectic exponentially fitted integrator. Comput. Phys. Comm. 174 (2006) 255-262
34. Stavroyiannis S., and Simos T.E. Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs. Appl. Numer. Math. 59 (2009) 2467-2474
35. Wambecq A. Rational Runge-Kutta methods for solving systems of ordinary differential equations. Computing 20 (1978) 333-342
36. Wynn P. Acceleration techniques for iterated vector and matrix. Math. Comp. 16 (1962) 301-322
37. Hairer E. Unconditionally stable explicit methods for parabolic equations. Numer. Math. 35 (1980) 57-68
38. Stiefel E., and Bettis D.G. Stabilization of Cowell's method. Numer. Math. 13 (1969) 154-175
39. Thomas R.M. Phase properties of high order almost P-stable formulae. BIT 24 (1984) 225-238
40. Franco J.M., and Palacios M. High-order P-stable multistep methods. J. Comput. Appl. Math. 30 (1990) 1-10