A nearly-conservative, high-order, forward Lagrange-Galerkin method for the resolution of scalar hyperbolic conservation laws

摘要

In this work, we present a novel Lagrange-Galerkin method for the resolution of scalar hyperbolic conservation laws. The scheme considers: (i) a conservative, weak, Lagrangian formulation which is formally discretized in space and in time with arbitrary order of accuracy, (ii) a forward-in-time integration of the fluid trajectories to allow for more stable and efficient time discretizations, (iii) nodal space-discretizations on unstructured triangular meshes and (iv) a novel and simple operator which employs the values of the fine-scale term of the solution to detect and capture the discontinuities. The method has been tested on several benchmark problems -including a hard case of non-convex flux- using a third-order time-integration formula and up to fourth-order finite elements, yielding the expected convergence rates both for smooth and discontinuous solutions. To the best of our knowledge, this is the first Lagrange-Galerkin method for hyperbolic conservation laws in the literature that allows for discontinuous solutions. (C) 2020 Elsevier B.V. All rights reserved.