This paper reviews Vlasov-based numerical methods used to model plasma in space physics and astrophysics. Plasma consists of collectively behaving charged particles that form the major part of baryonic matter in the Universe. Many concepts ranging from our own planetary environment to the Solar system and beyond can be understood in terms of kinetic plasma physics, represented by the Vlasov equation. We introduce the physical basis for the Vlasov system, and then outline the associated numerical methods that are typically used. A particular application of the Vlasov system is Vlasiator, the world’s first global hybrid-Vlasov simulation for the Earth’s magnetic domain, the magnetosphere. We introduce the design strategies for Vlasiator and outline its numerical concepts ranging from solvers to coupling schemes. We review Vlasiator’s parallelisation methods and introduce the used high-performance computing (HPC) techniques. A short review of verification, validation and physical results is included. The purpose of the paper is to present the Vlasov system and introduce an example implementation, and to illustrate that even with massive computational challenges, an accurate description of physics can be rewarding in itself and significantly advance our understanding. Upcoming supercomputing resources are making similar efforts feasible in other fields as well, making our design options relevant for others facing similar challenges.

For decades, monochromatic large‐scale ultralow frequency (ULF) waves with a period of about 30 s have been observed upstream of the quasi‐parallel bow shock. These waves typically propagate obliquely with respect to the interplanetary magnetic field (IMF), while the growth rate for the instability causing the waves is maximized parallel to the magnetic field. It has been suggested that the mechanism for the oblique propagation concerns wave refraction due to the spatial variability of the suprathermal ions, originating from the E × B drift component. We investigate the ULF foreshock under a quasi‐radial IMF with Vlasiator, which is a newly developed global hybrid‐Vlasov simulation solving the Vlasov equation for protons, while electrons are treated as a charge‐neutralizing fluid. We observe the generation of the 30 s ULF waves and compare their properties to previous literature and multipoint Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft observations. We find that Vlasiator reproduces the foreshock ULF waves in all reported observational aspects. We conclude that the variability of the density and velocity of the reflected back streaming ions determines the large‐scale structure of the foreshock, which affects the wave frequency, wavelength, and oblique propagation. We conclude that the wave refraction may also be at work for radial IMF conditions, which has earlier been thought of as an exception to the refraction mechanism due to the small E × B drift component. We suggest that additional refraction may be caused by the large‐scale spatial variability of the density and velocity of the back streaming ions.

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