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