On the consistency and convergence of classical Richardson extrapolation as applied to explicit one-step methods
Abstract
The consistency of the classical Richardson extrapolation (CRE), a simple and robust computational device, is analysed for the case where the underlying method is an explicit one-step numerical method for ordinary differential equations with order of consistency one or two. It is shown in the classical framework that the CRE increases the order of consistency by one. The convergence of the method is proved by the assumption that the time-stepping operator of the base method has the Lipschitz property in its second argument.
Keyword : consistency, convergence, explicit method, Richardson extrapolation, Taylor series expansion
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
K. Burrage. Parallel and Sequential Methods for Ordinary Differential Equations. Oxford University Press, Oxford, New York, 1992.
J.C. Butcher. Numerical Methods for Ordinary Differential Equations. Second edition, Wiley, New York, 2003. https://doi.org/10.1002/0470868279
I. Faragó, A. Havasi and Z. Zlatev. The convergence of explicit Runge-Kutta´ methods combined with Richardson extrapolation. In J. Brandts, J. Chleboun, S. Korotov, K. Segeth, J. Sistek and T. Vejchodsky(Eds.), Applications of Mathematics 2012, Navier-Stokes Equations and Related Nonlinear Problems, pp. 99–106, Prague, 2012. Institute of Mathematics AS CR. Available from Internet: http://eudml.org/doc/287853.
A. Gorgey. Theoretical analysis of symmetric Runge–Kutta methods. Malaysian Journal of Mathematical Sciences 12(3): 369–382, 2018.
E. Hairer, C. Lubich and G. Wanner. Geometric numerical integration: structurepreserving algorithms, 2nd edition. Springer-Verlag, Berlin, 2006.
E. Hairer, S.P. Nørsett and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, Berlin, 1987.
E. Hairer and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1991. https://doi.org/10.1007/978-3-662-09947-6
W. Hundsdorfer and J.G. Verwer. Numerical Method for Ordinary Differential Systems: The Initial Value Problem. John Wiley & Sons Ltd., New York, 1991.
J.D. Lambert. Numerical Methods for Ordinary Differential Equations. SpringerVerlag, Berlin, 2003.
L.F. Richardson. The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London.Series ASeries A, Containing Papers of a Mathematical or Physical Character, 210(5):307–357, 1911. https://doi.org/10.1098/rsta.1911.0009
L.F. Richardson. The deferred approach to the limit I. single lattice. Philosophical Transactions of the Royal Society of London. Series A, pp. 299–349, 1927.
E. Süli. Numerical Solution of Ordinary Differential Equations. University of Oxford, 2014.
Z. Zlatev, I. Dimov, I. Faragó, K. Georgiev, Á. Havasi and Tz. Ostromsky. Application of Richardson extrapolation for multi-dimensional advection equations, computers and mathematics with applications. Computers and Mathematics with Applications, 67(12):2279–2293, 2014. https://doi.org/10.1016/j.camwa.2014.02.028
Z. Zlatev, I. Dimov, I. Faragó and Á. Havasi. Richardson Extrapolation - Practical Aspects and Applications. De Gruyter, 2017.
Z. Zlatev, I. Faragó and Á. Havasi. Stability of the Richardson extrapolation together with the θ-method. Journal of Computational and Applied Mathematics, 235:507–517, 2010. https://doi.org/10.1016/j.cam.2010.05.052