2009, English, Article edition: Explicit Runge--Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems Burman, Erik; Ern, Alexandre; Fernández, Miguel Ángel

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Physical Description
  • text
Published
  • HAL - CCSD
  • 2009
Language
  • English

Edition details

Title
  • Explicit Runge--Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems
Author
  • Burman, Erik
  • Ern, Alexandre
  • Fernández, Miguel Ángel
Published
  • HAL - CCSD
  • 2009
Physical Description
  • text
Subjects
Notes
  • We analyze explicit Runge--Kutta schemes in time combined with stabilized finite elements in space to approximate evolution problems with a first-order linear differential operator in space of Friedrichs-type. For the time discretization, we consider explicit second- and third-order Runge--Kutta schemes. We identify a general set of properties on the spatial stabilization, encompassing continuous and discontinuous finite elements, under which we prove stability estimates using energy arguments. Then, we establish $L^2$-norm error estimates with (quasi-)optimal convergence rates for smooth solutions in space and time. These results hold under the usual CFL condition for third-order Runge--Kutta schemes and any polynomial degree in space and for second-order Runge--Kutta schemes and first-order polynomials in space. For second-order Runge--Kutta schemes and higher polynomial degrees in space, a tightened 4/​3-CFL condition is required. Numerical results are presented for the advection and wave equations.
  • HAL:hal-00380659, version 1
  • HAL:http:/​/​hal.archives-ouvertes.fr/​hal-00380659/​en/​
Language
  • English
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