One-step explicit methods for the numerical integration of perturbed oscillators

References

• Brusa , L. and Nigro , L. 1980 . A one-step method for direct integration of structural dynamic equations . Internat. J. Numer. Methods Eng , 15 : 685 – 699 .
   Web of Science ®Google Scholar
• Fang , Y. , Ming , Q. and Wu , X. 2010 . Extended RKN-type methods with minimal dispersion error for perturbed oscillators . Comput. Phys. Comm , 181 : 639 – 650 .
   Google Scholar
• Franco , J. M. 2003 . A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators . J. Comput. Appl. Math , 161 : 283 – 293 .
   Google Scholar
• Franco , J. M. , Gomez , I. and Randez , L. 2001 . Four-stage syplectic and P-stable SDIRKN methods with dispersion of high order . Numer. Algorithms , 26 : 347 – 363 .
   Google Scholar
• Hairer , E. 1980 . Unconditionally stable explicit methods for parabolic equations . Numer. Math , 35 : 57 – 68 .
   Google Scholar
• Hairer , E. , Lubich , Ch. and Wanner , G. 2002 . Numerical Geometric Integration , Berlin : Springer .
   Google Scholar
• Ixaru , Gr. and Vanden , Berghe G. 2004 . Exponential Fitting , Dortrecht : Kluwer Academic Publishers .
   Google Scholar
• Lambert , J. D. 1973 . Computational Methods in Ordinary Differential Equations , London : John Wiley & Sons Ltd .
   Google Scholar
• Lambert , J. D. and Watson , I. A. 1976 . Symmetric multistep methods for periodic initial value problems . J. Inst. Math. Appl , 18 : 189 – 202 .
   Google Scholar
• Li , Q. H. 2010 . Two-step explicit methods for second-order IVPs with oscillatory solutions . Int. J. Comput. Math , 87 ( 2 ) : 353 – 366 .
   Web of Science ®Google Scholar
• Li , Q. H. and Song , Y. Z. 2006 . Explicit one-step P-stable methods for second order periodic initial value problems . Numer. Math. J. Chinese Univ , 15 : 237 – 247 .
   Google Scholar
• Li , Q. H. and Wu , X. Y. 2004 . A two-step explicit P-stable method of high phase-lag order for second order initial value problems . Appl. Math. Comput , 151 : 17 – 26 .
   Google Scholar
• Simos , T. E. 1999 . Dissipative high phase-lag order Numerov-type methods for the numerical solution of the Schrödinger equation . Comput. Chem , 23 : 439 – 446 .
   Google Scholar
• Simos , T. E. and Tsitouras , Ch. 1997 . A P-stable eighth-order method for the numerical integration of periodic initial value problems . J. Comput. Phys , 130 : 123 – 128 .
   Google Scholar
• Simos , T. E. and Williams , P. S. 1999 . On finite difference methods for the solution of the Schrödinger equation . Comput. Chem , 23 : 513 – 554 .
   Google Scholar
• Simos , T. E. and Williams , P. S. 2001 . New insights in the development of Numerov-type methods with minimal phase-lag for the numerical solution of the Schrödinger equation . Comput. Chem , 25 : 77 – 82 .
   PubMedGoogle Scholar
• Sottas , G. 1984 . Rational Runge–Kutta methods are not suitable for stiff systems of ODEs . J. Comput. Appl. Math , 10 : 169 – 174 .
   Google Scholar
• Wambecq , A. 1978 . Rational Runge–Kutta method for solving systems of ordinary differential equations . Computing , 20 : 333 – 342 .
   Google Scholar
• Wu , X. Y. and Xia , J. L. 2000 . The vector form of a sixth-order A-stable explicit one-step method for stiff problems . Comput. Math. Appl , 39 : 247 – 257 .
   Google Scholar
• Yang , H. L. and Wu , X. Y. 2008 . Trigonometrically-fitted ARKN methods for perturbed oscillators . Appl. Numer. Math , 58 : 1375 – 1395 .
   Google Scholar
• Yang , H. L. , Wu , X. Y. , You , X. and Fang , Y. L. 2009 . Extended RKN-type methods for numerical integration of perturbed oscillators . Comput. Phys. Comm , 180 : 1777 – 1794 .
   Google Scholar
• van Den Berghe , G. , De Meyer , H. , Van Daele , M. and Van Hecke , T. 2000 . Exponentially-fitted Runge–Kutta methods . J. Comput. Appl. Math , 125 : 107 – 115 .
   Google Scholar
• van Der Houwen , P. J. and Sommeijer , B. P. 1987 . Explicit Runge–Kutta(–Nyström) methods with reduced phase errors for computing oscillating solutions . SIAM J. Numer. Anal , 24 : 595 – 617 .
   Google Scholar
• van Der Houwen , P. J. and Sommeijer , B. P. 1989 . Diagonally implicit Runge–Kutta–Nyström methods for oscillatory problems . SIAM J. Numer. Anal , 26 : 414 – 429 .
   Google Scholar