A robust high-order Lagrange-projection like scheme with large time steps for the isentropic Euler equations
This work provides a high-order numerical scheme for compressible flow simulations that maintains stability with large time steps, benefiting computational fluid dynamics practitioners.
The authors extend a first-order Lagrange-projection method to high-order for isentropic Euler equations, enabling large time steps by relaxing acoustic wave constraints. They prove positivity and entropy conditions for arbitrary spatial order and demonstrate stability and robustness in numerical experiments.
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 {\it 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 satisfy an entropy inequality for the numerical solution 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, though it allows the use of large time steps.