Implementation of symmetrizers in ordinary differential equations

Abstract

Two-step symmetrizers for the implicit midpoint and trapezoidal rules provide an alternative to the one-step smoothing formula for solving stiff ordinary differential equations. When used with the basic symmetric methods, these \(L\)-stable methods preserve the asymptotic error expansion in even powers of the step size and provide the necessary damping of oscillatory solutions. These new symmetrizers show effects similar to one-step smoothing but with the advantage of being order two. When generalized to higher order symmetric methods, such as the two-stage Gauss or the three-stage Lobatto IIIA, these symmetrizers can suppress order reduction for stiff problems. Here, we discuss one-step and two-step symmetrizers and their application in ordinary differential equations. We present numerical results with constant and variable step sizes that show the advantages of two-step symmetrizers over one-step symmetrizers of the implicit trapezoidal rule for stiff linear and nonlinear problems. References

• R. Bulirsch and J. Stoer, Numerical treatment of ordinary differential equations by extrapolation method, Numer. Math. 8(1):1–13, 1966. doi:10.1007/BF02165234
• R. P. K. Chan, Extrapolation of Runge–Kutta methods for stiff initial value problems. Thesis submitted for the degree of Doctor of Philosophy at the University of Auckland, 1989. https://researchspace.auckland.ac.nz/handle/2292/1842
• R. P. K. Chan, A-stability of implicit Runge–Kutta extrapolations, App. Numer. Math. 22(1–3):179–203, 1996. doi:10.1016/S0168-9274(96)00031-1
• R. P. K. Chan and N. Razali. Smoothing Effects on the IMR and ITR, Numer. Algorithms 65(3):401–420, 2013, doi:10.1007/s11075-013-9779-7
• C. F. Curtiss, and J. O. Hirschfelder, Integration of stiff equations, P. Natl. Acad. Sci. 38:235–243, 1952. doi:10.1073/pnas.38.3.235
• G. Dahlquist and B. Lindberg, On some implicit one-step methods for stiff differential equations, Royal Institute of Technology, 1973.
• H. T. Davis, Introduction to nonlinear differential and integral equations, Dover, New York, 1962. http://catalog.hathitrust.org/Record/000619023
• R. Frank, J. Schneid and C. W. Ueberhuber. Order results for implicit Runge–Kutta methods applied to stiff systems, SIAM. J. Numer. Analysis, 22(3):515–534, 1985. doi:10.1137/0722031
• A. Gorgey, Extrapolation of Symmetrized Runge-Kutta Methods. Thesis submitted for the degree of Doctor of Philosophy at the University of Auckland, 2012. https://researchspace.auckland.ac.nz/handle/2292/18996
• W. B. Gragg, On extrapolation algorithm for ordinary initial value problems, SIAM J. Numer. Anal. 2(3):384–403 1965. doi:10.1137/0702030
• E. Hairer and G. Wanner, Solving ordinary differential equations, II. Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1991. doi:10.1007/978-3-662-09947-6
• R. Holsapple, R. Iyer and D. Doman, Variable step-size selection methods for implicit integration schemes for ODEs, Int. J. Numer. Anal. Mod., 4(2):210–240, 2007. http://www.math.ualberta.ca/ijnam/Volume-4-2007/No-2-07/2007-02-04.pdf
• B. Lindberg, On the smoothing and extrapolation for the trapezoidal rule, BIT, 11(1):29–52, 1971. doi:10.1007/BF01935326
• A. Prothero and A. Robinson, On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comput. 28:145–162, 1974. doi:10.1090/S0025-5718-1974-0331793-2
• L. F. Richardson, J. A. Gaunt. The Deferred Approach to the Limit, Philos. T. R. Soc. A 226:299–361, 1927. doi:10.1098/rsta.1927.0008
• H. J. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin, 1973. http://www.springer.com/us/book/9783642654732