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\title{Using Butcher's Tables in Transient PDE Solvers}
\tocauthor{P. Solin} \author{} \institute{}
{\large \underline{Pavel Solin}}\\
University of Nevada, Reno\\
\\ \vspace{4mm}{\large Lukas Korous}\\
FEMhub Inc.\\
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 ( and they are
reproducible in the Networked Computing Laboratory (NCLab) at
{\sc J. C. Butcher}. {Numerical Methods for Ordinary Differential Equations}. J. Wiley \& Sons, 2003.
{\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.
{\sc P. Solin. K. Segeth and I. Dolezel}. {Higher-Order Finite Element Methods}. CRC Press, 2004.