Explicit pseudo two-step runge-kutta methods for parallel computersFootnote^∗

^∗This work was partly supported by DAAD, N.R.P.F.S. and QG-96-02

References

• Abramowitz , M. and Stegun , I.A. 1970 . “ Handbook of Mathematical Functions ” . In National Bureau of Standards Applied Mathematical Series , Vol. 55 , New York : Dover .
   Google Scholar
• Burrage , K. 1993 . Efficient block predictor-corrector methods with a small number of corrections . J. Comput. AppL Math , 45 : 139 – 150 .
   Google Scholar
• Burrage , K. 1995 . Parallel and Sequential Methods for Ordinary Differential Equations , Oxford : Clarendon Press .
   Google Scholar
• Butcher , J.C. 1987 . “ Runge- Kutta and General Linear Method ” . In The Numerical Analysis of Ordinary Differential Equations , New York : Wiley .
   Google Scholar
• Oiu , M.T. and Hamilton , H. 1987 . Parallel solution of ODE’s by multi-block methods . SIAM J. Sci. Statist. Comput , 8 : 137 – 157 .
   Google Scholar
• Cong , N.H. 1994 . Parallel iteration of symmetric Runge-Kutta methods for nonstut initial-value problems . J. Comput AppL Math , 51 : 117 – 125 .
   Google Scholar
• Cong , N.H. 1995 . Direct collocation-based two-step Runge-Kutta-Nyström methods . SEA Bull Math , 19 : 49 – 58 .
   Google Scholar
• Cong , N.H. 1996 . Explicit parallel two-step Runge-Kutta-Nystrdm methods . Computers Math. Applic , 32 : 119 – 130 .
   Google Scholar
• Cong N. H. Continuous and variable stepsize explicit pseudo two-step Runge-Kuttta methods, submitted for publication
   Google Scholar
• Cong , N.H. and Mitsui , T. 1996 . Collocation-based two-step Runge-Kutta methods . Japan J. Indust. Appl. Math , 13 : 171 – 183 .
   Google Scholar
• Cong , N.H. and Mitsui , T. 1997 . A class of explicit parallel two-step Runge-Kutta methods . Japan J. Indust. Appl. Math , 14 : 303 – 313 .
   Google Scholar
• Cong N. H. Strehmel K. Weiner R. A general class of explicit pseudo two-step RKN methods on parallel computers, submitted for publication
   Google Scholar
• Curtis , A.R. 1975 . High-order explicit Runge-Kutta formulae, their uses, and limitations . J. Inst Math. Appl , 16 : 35 – 55 .
   Google Scholar
• Hairer , E. 1978 . A Runge-Kutta method of order 10 . J. Inst. Math. Appl , 21 : 47 – 59 .
   Google Scholar
• Hairer , E. , Sett , S.P. and Wanner , G. 1987 . “ Solving Ordinary Differential Equations ” . In J. Nonstiff Problems , Berlin : Springer-Verlag .
   Google Scholar
• Van Der Houwen , P.J. and Cong , N.H. 1993 . Parallel block predictor-corrector methods of Runge-Kutta type . Appl Numer. Math , 13 : 109 – 123 .
   Google Scholar
• Van Der Houwen , P.J. and Sommeijer , B.P. 1990 . Parallel iteration of high-order Runge-Kutta methods with stepsize control . J. Comput. Appl. Math , 29 : 111 – 127 .
   Google Scholar
• Van Der Houwen , P.J. and Sommeijer , B.P. 1992 . Block Runge-Kutta methods on parallel computers . Z. Angew. Math. Mech , 68 : 3 – 127 .
   Google Scholar
• Hull , T.E. , Enright , W.H. , Fellen , B.M. and Sedgwick , A.E. 1972 . Comparing numerical methods for ordinary differential equations . SIAM J. Numer. Anal , 9 : 603 – 637 .
   Web of Science ®Google Scholar
• Jackiewizc , Z. , Renaut , R. and Feldstein , A. 1991 . Two-step Runge-Kutta methods . SIAMS J. Numer. Anal , 23 : 1165 – 1182 .
   Google Scholar
• Jackson K. R. Norsett S. P. The potential for parallelism in Runge-Kutta methods Part I: RK formulas in standard form, Tech. Rep. 239/90, Comp. Sci. Dept. Univ. Toronto, Canada
   Google Scholar
• Jackson K. R. Narsett S. P. The potential for parallelism in Runge-Kutta methods Part 2: RK predictor-corrector formulas, Tech. Rep., Comp. Sci. Dept., Univ. Toronto Canada 1991
   Google Scholar
• Lie , I. 1987 . “ Some aspects of parallel Runge-Kutta methods ” . In Division Numerical Mathematics , Norway : University of Trondheim . Report No. 3/87
   Google Scholar
• N⊘rsett , S.P. and Simonsen , H.H. . Numerical Methods for Ordinary Differential Equations . Proceedings L’Aquilla . 1987 . Edited by: Bellen , A. , Gear , C.W. and Russo , E. Berlin : Springer . Lecture Notes in Mathematics 1386
   Google Scholar