Limitation strategies for high-order discontinuous Galerkin schemes applied to an Eulerian model of polydisperse sprays

In this paper, we tackle the modeling and numerical simulation of polydisperse sprays. Starting from a kinetic description for point particles, we focus on an Eulerian high-order geometric method of moment (GeoMOM) in size and consider a system of partial differential equations on a vector of successive fractional size moments of order 0 to N/2, N > 2, over a compact size interval. These moments correspond to physical quantities, which can be interpreted in terms of the geometry of the interface at small scale. There exists a stumbling block for the usual approaches using high-order moment methods resolved with high-order numerical methods: the transport algorithm does not naturally preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells can create N-dimensional vectors which can fail to be moment vectors. We thus propose a new approach, as well as an algorithm, which is high-order in space with limited numerical diffusion, including at the boundaries of the state space, where a specific study is proposed. The main contribution of this work is the design and analysis of a high-order scheme preserving the bounds on the velocity, the moment space and capturing void and δ-shocks solutions. We show that such an approach is competitive compared to second order finite volume schemes, where limiters generate numerical diffusion and clipping at extrema. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. We focus in this paper on cartesian meshes and 2D test cases are presented where the accuracy and efficiency of the approach are assessed.