Thirteen ways to estimate global error

SummaryVarious techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations, particularly ordinary differential equations, are classified, described, and compared. For most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated.

[1]  L. Fox,et al.  Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations , 1947, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[2]  John Todd,et al.  Solution of Differential Equations by Recurrence Relations , 1950 .

[3]  Notes on Numerical Analysis--3: Solution of Differential Equations by Recurrence Relations , 1951 .

[4]  V. Pereyra On improving an approximate solution of a functional equation by deferred corrections , 1966 .

[5]  N. S. Barnett,et al.  Private communication , 1969 .

[6]  J. Butcher The effective order of Runge-Kutta methods , 1969 .

[7]  D. C. Joyce Survey of Extrapolation Processes in Numerical Analysis , 1971 .

[8]  M. N. Spijker On the structure of error estimates for finite-difference methods , 1971 .

[9]  Hans J. Stetter,et al.  Local Estimation of the Global Discretization Error , 1971 .

[10]  J. M. Watt Numerical Initial Value Problems in Ordinary Differential Equations , 1972 .

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

[12]  Hans J. Stetter Economical Global Error Estimation , 1974 .

[13]  Runge-Kutta Methods with Global Error Estimates , 1975 .

[14]  The method of iterated defect-correction and its application to two-point boundary value problems , 1975 .

[15]  L. Shampine,et al.  Computer solution of ordinary differential equations : the initial value problem , 1975 .

[16]  S. Scholz,et al.  H. J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations. (Springer Tracts in Natural Philosophy. Ed. Coleman, B. D., Band 23). XVI + 388 S. m. 12 Fig. Berlin/Heidelberg/New York 1973. Springer‐Verlag. Preis geb. DM 120,— , 1975 .

[17]  Alan C. Hindmarsh,et al.  A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations , 1975, TOMS.

[18]  K. Böhmer A defect correction method for functional equations , 1976 .

[19]  P. Zadunaisky On the estimation of errors propagated in the numerical integration of ordinary differential equations , 1976 .

[20]  Lawrence F. Shampine,et al.  Global Error Estimates for Ordinary Differential Equations , 1976, TOMS.

[21]  Reinhard Frank,et al.  The method of Iterated Defect-Correction and its application to two-point boundary value problems , 1976 .

[22]  H. Stetter The defect correction principle and discretization methods , 1978 .

[23]  H. J. Stetter,et al.  Global error estimation in ODE-solvers , 1978 .

[24]  Peter C. C. Wang,et al.  Information linkage between applied mathematics and industry , 1979 .

[25]  Ian Gladwell,et al.  Initial Value Routines in the NAG Library , 1979, TOMS.

[26]  Computational Experiments on Two Error Estimation Procedures for Ordinary Differential Equations. , 1979 .

[27]  D. L. Hicks,et al.  COMPARISON BETWEEN TWO ERROR ESTIMATION PROCEDURES , 1979 .

[28]  P. M. Dew,et al.  Estimating and controlling the global error in Gear's method , 1979 .

[29]  Global error estimation for the implicit trapezoidal rule , 1980 .

[30]  Bengt Lindberg Error estimation and iterative improvement for discretization algorithms , 1980 .

[31]  Klaus Böhmer,et al.  Discrete Newton methods and iterated defect corrections , 1981 .

[32]  Long Chen INTRODUCTION TO MULTIGRID METHODS , 2005 .

[33]  Klaus Böhmer,et al.  Discrete correction methods for operator equations , 1981 .

[34]  J. G. Verwer,et al.  ESTIMATING THE GLOBAL ERROR OF RUNGE-KUTTA APPROXIMATIONS FOR ORDINARY DIFFERENTIAL EQUATIONS , 1982 .

[35]  Global error estimation in ordinary initial value problems , 1982 .

[36]  Robert D. Skeel,et al.  A Theoretical Framework for Proving Accuracy Results for Deferred Corrections , 1982 .

[37]  Germund Dahlquist On the control of the global error in stiff initial value problems , 1982 .

[38]  A. Brandt Guide to multigrid development , 1982 .

[39]  Lawrence F. Shampine,et al.  Global error estimation for stiff ODEs , 1984 .

[40]  Asymptotic theory of the global error and some techniques of error estimation , 1984 .

[41]  K. Bohmer Defect Correction Methods: Theory and Applications , 1984 .

[42]  Klaus Böhmer Defect correction methods - theory and applications , 1984, Computing : Supplementum.

[43]  Hans J. Stetter,et al.  The Defect Correction Approach , 1984 .

[44]  Lawrence F. Shampine,et al.  Global Error Estimates for ODEs Based on Extrapolation Methods , 1985 .