Towards a compact high-order method for non-linear hyperbolic systems. I: The Hermite Least-Square Monotone (HLSM) reconstruction

Towards a compact high-order method for non-linear hyperbolic systems, II. The Hermite-HLLC scheme

“…If we consider (8) or (9) with sufficiently smooth data, the exact solution can be obtained from (12). Indeed, the characteristic passing through any given point x 0 x(t = 0) on the x-axis, is…”

“…Consequently, the linear component, v, of the solution, becomes singular. In order to solve the Cauchy problems in those ''no classical" situations, it is necessary to introduce new singularities called ''d-shocks", which are solution of (8) or (9). More precisely, one can demonstrate that the solution of the Riemann problem (9), is the following one [5]:…”

“…This procedure aims at generating a compact high-order numerical method for systems of hyperbolic conservation laws [9]. For that purpose, we defined as discrete unknowns not only the primitive variable, u, but also its first spatial derivative, v(u x ).…”

“…First, we demonstrate that the extended version of the HLLE Riemann solver is compatible with the existence of d-shocks into the numerical solution. Then, we modify the HLSM method [9], in order to capture solutions that present a Dirac measure. This point is achieved by adding a new constraint to the numerical least-square system.…”

“…In [9] we developed a new least-square interpolation that relies upon the cell-averages of the variable and its first spatial derivatives.…”