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We first study the one-dimensional dusty gas flow modeled by the two-phase system composed of a gaseous carrier (gas phase) and a particulate suspended phase (dust phase). The gas phase is modeled by the compressible Euler equations of gas dynamics and the dust phase is modeled by the pressureless gas dynamics equations. These two sets of conservation laws are coupled through source terms that model momentum and heat transfers between the phases. When an Eulerian method is adopted for this model, one can notice the obtained numerical results are typically significantly affected by numerical diffusion. This phenomenon occurs since the pressureless gas equations are nonstrictly hyperbolic and have degenerate structure in which singular delta shocks are formed, and these strong singularities are vulnerable to the numerical diffusion. We introduce a low dissipative hybrid finite-volume-particle method in which the compressible Euler equations for the gas phase are solved by a central-upwind scheme, while the pressureless gas dynamics equations for the dust phase are solved by a sticky particle method. The obtained numerical results demonstrate that our hybrid method provides a sharp resolution even when a relatively small number of particle is used. We then extend the hybrid finite-volume-particle method to the three-dimensional dusty gas flows with axial symmetry. In the studied model, gravitational effects are taken into account. This brings an additional level of complexity to the development of the finite-volume-particle method since a delicate balance between the flux and gravitational source terms should be respected at the discrete level. We test the proposed method on a number of numerical examples including the one that models volcanic eruptions. Math. classification. 65M08, 76M12, 76M28, 86-08, 76M25, 35L65.

Hybrid Finite-Volume-Particle Method for Dusty Gas Flows