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Article Dans Une Revue SIAM Journal on Numerical Analysis Année : 2010

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

Résumé

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.
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Dates et versions

hal-00380659 , version 1 (04-05-2009)

Identifiants

Citer

Erik Burman, Alexandre Ern, Miguel Angel Fernández. Explicit Runge--Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems. SIAM Journal on Numerical Analysis, 2010, 48 (6), pp.2019-2042. ⟨10.1137/090757940⟩. ⟨hal-00380659⟩
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