Stability and accuracy of time discretizations for initial value problems

SummaryThis paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.

[1]  G. Dahlquist.  Convergence and stability in the numerical integration of ordinary differential equations , 1956 .

[2]  P. Henrici.  Discrete Variable Methods in Ordinary Differential Equations , 1962 .

[3]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[4]  E. Cheney.  Introduction to approximation theory , 1966 .

[5]  J. D. Buckholtz,et al.  A Characterization of the Exponential Series , 1966 .

[6]  Olof B. Widlund,et al.  A note on unconditionally stable linear multistep methods , 1967 .

[7]  C. W. Gear,et al.  The automatic integration of stiff ordinary differential equations , 1968, IFIP Congress.

[8]  C. W. Gear,et al.  Numerical initial value problem~ in ordinary differential eqttations , 1971 .

[9]  Y. Genin.  A new approach to the synthesis of stiffly stable linear multistep formulas , 1973 .

[10]  Colin W. Cryer.  A new class of highly-stable methods:A0-stable methods , 1973 .

[11]  Rolf Jeltsch,et al.  Stiff Stability and Its Relation to $A_0 $- and $A(0)$-Stability , 1976 .

[12]  William F. Ames,et al.  3 – Elliptic equations , 1977 .

[13]  R. Jeltsch.  Complete characterization of multistep methods with an interval of periodicity for solving , 1978 .

[14]  E. Hairer,et al.  Order stars and stability theorems , 1978 .

[15]  J. M. Varah.  Stiffly Stable Linear Multistep Methods of Extended Order , 1978 .

[16]  G. Dahlquist.  On accuracy and unconditional stability of linear multistep methods for second order differential equations , 1978 .

[17]  R. Jeltsch.  Stability on the imaginary axis andA-stability of linear multistep methods , 1978 .

[18]  E. Hairer,et al.  WhenI-stability impliesA-stability , 1978 .

[19]  O. Østerby,et al.  Regions of Absolute Stability , 1979 .

[20]  E. Hairer.  Unconditionally stable methods for second order differential equations , 1979 .

[21]  Stability of linear multistep methods on the imaginary axis , 1981 .

[22]  O. Nevanlinna,et al.  Stability of explicit time discretizations for solving initial value problems , 1982 .