We developed a new numerical simulator of slurry flow in hydraulic fractures. It is based on a continuum model developed under the assumptions of lubrication theory. We consider slurry containing fluid and proppant components. Proppant particles settle under gravity and bridge in the flow domain. Fluid fluxes are calculated from an elliptic boundary value problem for given fracture geometry and sources/sinks distribution. We solve it using a conservative finite volume method and then apply empirical closure relations to get the velocity field for solids. The transport of slurry components is described by nonlinear hyperbolic equations with fluxes depending on proppant concentration and particle size/fracture width ratio. These equations admit solutions with concentration shock waves and rarefaction fans. They must be represented accurately in the computational algorithm. We utilize an Eulerian approach for the discretization of nonlinear proppant fluxes. After getting the fluxes on the interfaces between Eulerian grid cells we follow a Lagrangian approach to simulate the proppant transport along the given velocity field. In this study we discuss key implementation aspects: time step selection, regularization, fluxes discretization, seeding, reseeding, and movement of Lagrangian particles. We verify the simulator on benchmark solutions focused on advection, batch proppant settling, slurry slumping and fluid displacement by proppant.