An adaptive spectral element method for systems of conservation laws

We present a novel method for the numerical solution of systems of conservation laws. For the space discretization, the scheme considers high-order continuous finite elements stabilized via subgrid modeling, as well as highly anisotropic adaptive meshes in order to capture efficiently any sharp features in the solution. Time integration is carried out via a time-step adaptive, linearly implicit Runge–Kutta method, which allows large time steps and requires only the solution of a linear system of equations at each internal stage. An important characteristic of the present method is that the mesh is adapted before solving for the next time interval, and not afterwards as in the common procedure. Furthermore, the error arising from the inexact solution of the linear systems is also estimated and controlled, in such a way that the numerical solution is sufficiently accurate and the linear systems are not oversolved. Numerical experiments, including a computationally difficult case of non-convex flux and the Euler equations for compressible flows, were performed with up to eight-degree elements and a third-order time marching formula to demonstrate the capabilities of the method.