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New High‐Order Runge‐Kutta Formulas with an Arbitrarily Small Truncation Error
Author(s) -
Fehlberg Erwin
Publication year - 1966
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19660460102
Subject(s) - runge–kutta methods , truncation error , truncation (statistics) , mathematics , order (exchange) , reduction (mathematics) , term (time) , differential equation , zero (linguistics) , calculus (dental) , mathematical analysis , statistics , geometry , physics , linguistics , philosophy , finance , quantum mechanics , economics , medicine , dentistry
New high‐order Runge‐Kutta formulas for systems of first‐ and second‐order differential equations are derived. These new formulas, although similar to earlier formulas of the author, offer the advantage of a greatly reduced truncation error. By the proper choice of a parameter the leading term of the truncation error can be made arbitrarily small (but not zero) without increase of the overall computational expense per step, as compared with the earlier formulas. 24‐digit tables for the new Runge‐Kutta coefficients are presented for 8‐th through 12‐th order formulas. Two examples demonstrate the advantage of the new formulas as expressed in a substantial reduction of the number of required integration steps and, therefore, of the total computer running time as well.