Path-conservative in-cell discontinuous reconstruction schemes for non conservative hyperbolic systems

Abstract

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. We introduce the basics of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic, and apply it to a 2x2 non conservative toy model, and a 3x3 gas dynamics system in Lagrangian coordinates. The strategy allows in particular to compute exactly isolated shocks. Numerical evidences are proposed.

In-cell Discontinuous Reconstruction path-conservative methods for non conservative hyperbolic systems -- Second-order extension

“…Let us prove that isolated shock waves are exactly captured by the scheme and contain no spurious numerical diffusion. Although the proof is essentially the same as in [11], it is included for the sake of completeness.…”

“…The goal of this article is to extend the in-cell discontinuous reconstruction methods introduced in [11] to second-order accuracy. To do this, these numerical methods will be first written as high-order path-conservative schemes (see for example [6,10]) and then, depending of the smoothness of the numerical solution, a standard MUSCL-Hancock reconstruction (see [24] and [25]) or a discontinuous one is used in the cell to update the numerical solution.…”

In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension

“…The design of finite-difference or finite-volume methods satisfying these four properties is difficult in general. Nevertheless, different techniques have been introduced to overcome, at least partially, this convergence issue: [4], [3], [2], [5], [8], [12], [13], [15], [22], [11]. In particular the path-conservative entropy stable methods introduced in [8] and extended to DG high-order methods in [18] significantly reduce the convergence error: to do this, entropy-conservative numerical methods are first introduced that are stabilized by means of a discretization of the viscous term of the regularized equation (1.6).…”

“…More recently, in [11], an in-cell discontinuous reconstruction technique has been added to first-order path-conservative methods that allows one to capture correctly weak solutions with isolated shock waves.…”

“…The goal of this article is to extend the in-cell discontinuous reconstruction methods introduced in [11] to second-order accuracy. To do this, these numerical methods will be first written as high-order path-conservative schemes (see for example [6,10]) and then, depending of the smoothness of the numerical solution, a standard MUSCL-Hancock reconstruction (see [25] and [26]) or a discontinuous one is used in the cell to update the numerical solution.…”