论文信息 - On smoothing and order reduction effects for implicit Runge-Kutta formulae

On smoothing and order reduction effects for implicit Runge-Kutta formulae

Abstract It is well known that many important classes of Runge—Kutta methods suffer an order reduction phenomenon when applied to certain classes of stiff problems. In particular, the s-stage Gauss methods with stage order s and order of consistency 2s behave like methods of order s when applied to the class of singularly perturbed problems. In this paper we will show that the process of smoothing can ameliorate this effect, when dealing with initial-value problems, by first studying the effect of smoothing on the standard Prothero—Robinson problem and then by extending the analysis to the general class of singularly perturbed problems.

Kevin Burrage | R. P. K. Chan | K. Burrage | R. Chan

[1] P. Deuflhard,et al. A semi-implicit mid-point rule for stiff systems of ordinary differential equations , 1983 .

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

[3] J. R. E. O’Malley. On Nonlinear Singularly Perturbed Initial Value Problems , 1988 .

[4] Ernst Hairer,et al. The numerical solution of differential-algebraic systems by Runge-Kutta methods , 1989 .

[5] B. Lindberg. On smoothing and extrapolation for the trapezoidal rule , 1971 .

[6] Winfried Auzinger,et al. Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case , 1989 .

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

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

[9] W. Auzinger. ASYMPTOTIC ERROR EXPANSIONS FOR STIFF EQUATIONS . PART 1 : THE STRONGLY STIFF CASE , 2021 .

[10] William B. Gragg,et al. On Extrapolation Algorithms for Ordinary Initial Value Problems , 1965 .

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

[12] John C. Butcher,et al. On symmetrizers for Gauss methods , 1991 .