Burman, E;
Ern, A;
Fernandez, MA;
(2010)
Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems.
SIAM Journal on Numerical Analysis
, 48
(6)
pp. 2019-2042.
10.1137/090757940.
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Abstract
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 space 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 smooth and rough solutions. The case of finite volumes is briefly discussed.
| Type: | Article |
|---|---|
| Title: | Explicit Runge–Kutta Schemes and Finite Elements with Symmetric Stabilization for First-Order Linear PDE Systems |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1137/090757940 |
| Publisher version: | http://dx.doi.org/10.1137/090757940 |
| Language: | English |
| Additional information: | Copyright © 2010 Society for Industrial and Applied Mathematics |
| Keywords: | first-order PDEs, transient problems, stabilized finite elements, discontinuous Galerkin, explicit Runge-Kutta schemes, stability, convergence |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/1384729 |
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