A nearly-conservative, high-order, forward Lagrange–Galerkin method for the resolution of compressible flows on unstructured triangular meshes

In this work, we present a novel Lagrange–Galerkin method for the resolution of compressible and inviscid flows. The scheme considers: (i) high-order continuous space discretizations on unstructured triangular meshes, (ii) high-order implicit–explicit Runge-Kutta schemes for the time discretization, (iii) conservation of mass, momentum and total energy, as long as some integrals in the formulation are computed exactly, and (iv) subgrid-stabilization and discontinuity-capturing operators based on Brenner's model [51] (2006) for viscous flows. The method has been tested on several benchmark problems using a fourth-order time-marching formula and up to fifth-order continuous finite elements, yielding the expected results both for smooth and discontinuous solutions.