Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations

Runge-Kutta methods are studied when applied to stiff differential equations containing a small stiffness parameter ε. The coefficients in the expansion of the global error in powers of ε are the global errors of the Runge-Kutta method applied to a differential algebraic system. A study of these errors and of the remainder of the expansion yields sharp error bounds for the stiff problem. Numerical experiments confirm the results.

[1]  R. Alexander.  Diagonally implicit runge-kutta methods for stiff odes , 1977 .

[2]  F. Hoppensteadt.  Properties of solutions of ordinary differential equations with small parameters , 1971 .

[3]  R. Weiner,et al.  B-convergence results for linearly implicit one step methods , 1987 .

[4]  Olavi Nevanlinna,et al.  Matrix valued versions of a result of von Neumann with an application to time discretization , 1985 .

[5]  L. Segel,et al.  Introduction to Singular Perturbations. By R. E. O'MALLEY, JR. Academic Press, 1974. $ 16.50. , 1975, Journal of Fluid Mechanics.

[6]  Gerhard Wanner,et al.  A study of Rosenbrock-type methods of high order , 1981 .

[7]  Richard C Aiken,et al.  Stiff computation , 1985 .

[8]  Willard L. Miranker,et al.  Numerical Methods for Stiff Equations , 1980 .

[9]  Heinz-Otto Kreiss,et al.  Problems with different time scales , 1992, Acta Numerica.

[10]  C. W. Gear,et al.  Differential-algebraic equations index transformations , 1988 .

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

[12]  Lawrence F. Shampine,et al.  Implementation of Rosenbrock Methods , 1982, TOMS.

[13]  P. Rentrop,et al.  Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations , 1979 .

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

[15]  L. Petzold.  Order results for implicit Runge-Kutta methods applied to differential/algebraic systems , 1986 .

[16]  E. Griepentrog,et al.  Differential-algebraic equations and their numerical treatment , 1986 .

[17]  E. Hairer,et al.  Extrapolation at stiff differential equations , 1988 .

[18]  Willem Hundsdorfer,et al.  The order ofB-convergence of algebraically stable Runge-Kutta methods , 1987 .

[19]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[20]  M. Roche,et al.  Rosenbrock methods for Differential Algebraic Equations , 1987 .

[21]  Michel Roche,et al.  Implicit Runge-Kutta methods for differential algebraic equations , 1989 .

[22]  J. Neumann,et al.  Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Erhard Schmidt zum 75. Geburtstag in Verehrung gewidmet , 1950 .

[23]  Ernst Hairer,et al.  A note onD-stability , 1984 .

[24]  E. Hairer,et al.  On the stability of semi-implicit methods for ordinary differential equations , 1982 .

[25]  W. Hundsdorfer.  Stability andB-convergence of linearly implicit Runge-Kutta methods , 1986 .

[26]  P. G. Thomsen,et al.  Local error control inSDIRK-methods , 1986 .

[27]  P. Deuflhard,et al.  One-step and extrapolation methods for differential-algebraic systems , 1987 .