# hpfem/esco2012-boa

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 \title{Using Butcher's Tables in Transient PDE Solvers} \tocauthor{P. Solin} \author{} \institute{} \maketitle \begin{center} {\large \underline{Pavel Solin}}\\ University of Nevada, Reno\\ {\tt solin@unr.edu} \\ \vspace{4mm}{\large Lukas Korous}\\ FEMhub Inc.\\ {\tt lukas@femhub.com} \end{center} \section*{Abstract} We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combine adaptive higher-order finite elements ($hp$-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge-Kutta methods in time. Weak formulation is only created for the stationary residual, and the Runge-Kutta methods are specified via their Butcher's tables. Around 30 Butcher's tables for various Runge-Kutta methods with numerically verified orders of local and global truncation errors are provided. A time-dependent benchmark problem with known exact solution that contains a sharp moving front is introduced, and it is used to compare the quality of seven embedded implicit higher-order Runge-Kutta methods. Numerical experiments also include a comparison of adaptive low-order FEM and $hp$-FEM with dynamically changing meshes. All numerical results presented in this paper were obtained using the open source library Hermes (http://hpfem.org/hermes) and they are reproducible in the Networked Computing Laboratory (NCLab) at http://nclab.com. \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{ref1} {\sc J. C. Butcher}. {Numerical Methods for Ordinary Differential Equations}. J. Wiley \& Sons, 2003. \bibitem{ref2} {\sc P. Solin and L. Korous}. {Adaptive Higher-Order Finite Element Methods for Transient PDE Problems Based on }. Journal of Computational Physics, Volume 231, Issue 4, 20 February 2012, pp. 1635-1649. \bibitem{ref3} {\sc P. Solin. K. Segeth and I. Dolezel}. {Higher-Order Finite Element Methods}. CRC Press, 2004. \end{thebibliography}