Abstract
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.
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