TY - JOUR
T1 - Regularity properties of multistage integration methods
AU - Jackiewicz, Zdzislaw
AU - Vermiglio, R.
AU - Zennaro, M.
N1 - Funding Information:
* Corresponding author. 1 The work of the first author was supported by the National Science Foundation under grant NSF DMS-9208048 and by the Italian government. The work of the second and the third author was supported by the Italian M.U.R.S.T., fimds 40% and 60%.
PY - 1997/12/23
Y1 - 1997/12/23
N2 - The numerical method for ordinary differential equations is regular if it has the same set of finite asymptotic values as the underlying differential system. This paper examines the regularity and strong regularity properties of diagonally implicit multistage integration methods (DIMSIMs) introduced recently by J.C. Butcher. A sufficient condition for regularity and strong regularity of such methods of any order is given and it is proved that this condition is also necessary for two-step two-stage DIMSIMs of order greater than or equal to two. It is also demonstrated that there exist regular schemes in the class of explicit DIMSIMs. This is in contrast to explicit Runge-Kutta methods with more than one stage, which are always irregular.
AB - The numerical method for ordinary differential equations is regular if it has the same set of finite asymptotic values as the underlying differential system. This paper examines the regularity and strong regularity properties of diagonally implicit multistage integration methods (DIMSIMs) introduced recently by J.C. Butcher. A sufficient condition for regularity and strong regularity of such methods of any order is given and it is proved that this condition is also necessary for two-step two-stage DIMSIMs of order greater than or equal to two. It is also demonstrated that there exist regular schemes in the class of explicit DIMSIMs. This is in contrast to explicit Runge-Kutta methods with more than one stage, which are always irregular.
KW - Asymptotic values
KW - General linear method
KW - Ordinary differential equation
KW - Regularity
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U2 - 10.1016/S0377-0427(97)00194-5
DO - 10.1016/S0377-0427(97)00194-5
M3 - Article
AN - SCOPUS:0031378235
SN - 0377-0427
VL - 87
SP - 285
EP - 302
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 2
ER -