A robust high-order Lagrange-projection like scheme with large time steps for the isentropic Euler equations

Florent Renac (1)

1 ONERA - The French Aerospace Lab [Châtillon]

We present an extension to high-order of a first-order Lagrange-projection like method for the approximation of the Euler equations introduced in Coquel et al. (Math. Comput., 79 (2010), pp. 1493–1533). The method is based on a decomposition between acoustic and transport operators associated to an implicit-explicit time integration, thus relaxing the constraint of acoustic waves on the time step. We propose here to use a discontinuous Galerkin method for the space approximation. Considering the isentropic Euler equations, we derive conditions to keep positivity of the mean value of density and to satisfy a discrete entropy inequality in each element of the mesh at any approximation order in space. These results allow to design limiting procedures to restore these properties at nodal values within elements. Numerical experiments support the conclusions of the analysis and highlight stability and robustness of the present method, while it allows the use of large time steps.

Mots clés: isentropic Euler equation; positivity-preserving; entropy-satisfying; discontinuous Galerkin method; explicit-implicit; Lagrange-projection

Domaines: Mécanique des fluides [physics.class-ph]; Mathématiques [math]

Florent Renac. A robust high-order Lagrange-projection like scheme with large time steps for the isentropic Euler equations. Numerische Mathematik, 2017, 135 (2), pp.493-519. ⟨10.1007/s00211-016-0807-0⟩. ⟨hal-01851751⟩