B-convergence of the implicit midpoint rule and the trapezoidal rule

We present upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes. Specializing our results for the case of constant stepsizes they easily prove second order optimal B-convergence for both methods.1980 AMS Subject Classification: 65L05, 65L20.

[1]  G. Dahlquist.  Stability and error bounds in the numerical integration of ordinary differential equations , 1961 .

[2]  J. W. Schmidt.  G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. 85 S. Stockholm 1959. K. Tekniska Högskolans Handlingar , 1961 .

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

[4]  A. R. Gourlay.  A note on trapezoidal methods for the solution of initial value problems , 1970 .

[5]  J. Lambert.  Computational Methods in Ordinary Differential Equations , 1973 .

[6]  H. Stetter.  Analysis of Discretization Methods for Ordinary Differential Equations , 1973 .

[7]  A. Prothero,et al.  On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations , 1974 .

[8]  O. Nevanlinna.  On error bounds forG-stable methods , 1976 .

[9]  G. Dahlquist.  Error analysis for a class of methods for stiff non-linear initial value problems , 1976 .

[10]  Owe Axelsson.  Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form , 1977 .

[11]  Werner Liniger,et al.  Contractive methods for stiff differential equations Part II , 1978 .

[12]  K. Burrage,et al.  Non-linear stability of a general class of differential equation methods , 1980 .

[13]  Christoph W. Ueberhuber,et al.  The Concept of B-Convergence , 1981 .

[14]  Werner Liniger,et al.  Stability of Two-Step Methods for Variable Integration Steps , 1983 .

[15]  Owe Axelsson,et al.  Error estimates over infinite intervals of some discretizations of evolution equations , 1984 .

[16]  J. Verwer,et al.  Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations , 1984 .

[17]  W. Hundsdorfer,et al.  The numerical solution of nonlinear stiff initial value problems: an analysis of one step methods , 1985 .

[18]  Christoph W. Ueberhuber,et al.  Stability Properties of Implicit Runge–Kutta Methods , 1985 .

[19]  Christoph W. Ueberhuber,et al.  Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems , 1985 .