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Numerical analyses of the liquid fuel injection and subsequent fuel-air mixing for a high-tumble direct injection engine with an active pre-chamber ignition system at operation conditions of 2000 RPM are presented. The Navier-Stokes equations for compressible in-cylinder flow are solved numerically using a hierarchical Cartesian mesh based finite-volume method. To determine the fuel vapor before ignition large-eddy flow simulations are two-way coupled with the spray droplets in a Lagrangian Particle Tracking (LPT) formulation. The combined hierarchical Cartesian mesh ensures efficient usage of high performance computing systems through solution adaptive refinement and dynamic load balancing. Computational meshes with approximately 170 million cells and 1.0 million spray parcels are used for the simulations.