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Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems

journal contribution
posted on 2023-06-07, 22:50 authored by Erik Burman, Alexandre Ern, Miguel A Fernández
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

History

Publication status

  • Published

Journal

SIAM Journal on Numerical Analysis

ISSN

0036-1429

Publisher

Society for Industrial and Applied Mathematics

Issue

6

Volume

48

Page range

2019-2042

Pages

24.0

Department affiliated with

  • Mathematics Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2012-02-06

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