A Hermite upwind WENO scheme for solving hyperbolic conservation laws

This paper proposes a new WENO procedure to compute problems containing both discontinuities and a large disparity of characteristic scales. In a one-dimensional context, the WENO procedure is defined on a three-points stencil and designed to be sixth-order in regions of smoothness. We define a finite-volume discretization in which we consider the cell averages of the variable and its first derivative as discrete unknowns. The reconstruction of their point-values is then ensured by a unique sixth-order Hermite polynomial. This polynomial is considered as a symmetric and convex combination, by ideal weights, of three fourth-order polynomials: a central polynomial, defined on the three-points stencil, is combined with two polynomials based on the left and the right two-points stencils. The symmetric nature of such an interpolation has an important consequence: the choice of ideal weights has no influence on the properties of the discretization. This advantage enables to formulate the Hermite interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, nonlinear weights are then defined. To deal with the peculiarities of the Hermite interpolation near discontinuities, we define a new procedure in order for the nonlinear weights to smoothly evolve between the ideal weights, in regions of smoothness, and one-sided weights, otherwise. The resulting scheme is a sixth-order WENO method based on central Hermite interpolation and TVD Runge-Kutta time-integration. We call this scheme the HCWENO6 scheme. Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In these experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems.