By combining concepts from particle-in-cell (PIC) and hybridized discontinuous Galerkin (HDG) methods, we present a particle–mesh scheme for flow and transport problems which allows for diffusion-free advection while satisfying mass and momentum conservation – locally and globally – and extending to high-order spatial accuracy. This is achieved via the introduction of a novel particle–mesh projection operator which casts the particle–mesh data transfer as a PDE-constrained optimization problem, permitting advective flux functionals at cell boundaries to be inferred from particle trajectories. This optimization problem seamlessly fits in a HDG framework, whereby the control variables in the optimization problem serve as advective fluxes in the HDG scheme. The resulting algebraic problem can be solved efficiently using static condensation. The performance of the scheme is demonstrated by means of numerical examples for the linear advection–diffusion equation and the incompressible Navier–Stokes equations. The results demonstrate optimal spatial accuracy, and when combined with a θ time integration scheme, second-order temporal accuracy is shown.@en

A particle-mesh strategy is presented for scalar transport problems which provides diffusion-free advection, conserves mass locally (i.e. cellwise) and exhibits optimal convergence on arbitrary polyhedral meshes. This is achieved by expressing the convective field naturally located on the Lagrangian particles as a mesh quantity by formulating a dedicated particle-mesh projection based via a PDE-constrained optimization problem. Optimal convergence and local conservation are demonstrated for a benchmark test, and the application of the scheme to mass conservative density tracking is illustrated for the Rayleigh–Taylor instability.@en