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__Two-Step Symmetrization with Extrapolation__

N. Razali and R.P.K. Chan

**Abstract**

The use of Richardson extrapolation in accelerating convergence of a sequence is more efficient if the method has the $h^2$-asymptotic error expansion. Gragg first proved the existence of this property for the explicit midpoint rule in ordinary differential equations. He introduced the concept of smoothing to suppress the oscillatory parasitic component of the numerical solution and showed the advantage of smoothing applied with extrapolation [2]. In [1], this concept is generalized to arbitrary symmetric Runge-Kutta methods and called symmetrization. The two-step symmetrizer for the implicit midpoint and trapezoidal rules was constructed in [3]. In this talk, we discuss the advantages of using a one-step symmetrizer and investigate what other improvements there are in using two-step symmetrizers. We present experimental results that show the two-step symmetrizer can be more accurate and efficient in certain cases.

**Bibliography**

[1] R.P.K. Chan, Extrapolation of Runge-Kutta Methods for Stiff Initial Value Problems. PhD Thesis. University of Auckland, 1989.

[2] W.B. Gragg, On Extrapolation Algorithms for Ordinary Initial Value Problems, SIAM J. Numer. Anal. 2 (1965), pp. 384-403.

[3] N. Razali and R.P.K. Chan, Symmetrizers for Runge-Kutta Methods, Proceedings of International Conference on Mathematical Sciences and Statistics, ICMSS 2013, (2013). Preprint.