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Stability of Explicit Runge-Kutta Methods for High Order Finite Element Approximation of Linear Parabolic Equations

  • Weizhang Huang
  • Lennard Kamenski
  • Jens Lang
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.Department of MathematicsGraduate School of Computational Engineering, and Center of Smart Interfaces, Technische Universität DarmstadtDarmstadtGermany

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