# Modeling of Geophysical Flows through GPUFOW
## 1 Introduction  
Computational fluid dynamics (CFD), the art of modeling fluid flows by numerically solving the differential equations that describe their behavior, is a powerful method in the field of scientific research to expand our understanding of the physical laws of nature and how they apply to the world that surrounds us (for example, to better understand which rheological model best describes the behavior of lava flows and how this affects their emplacement), and in practical applications (e.g., for the forecasting of lava flow paths). Land use planning and natural risk mitigation, for example, benefit from hazard maps and scenario forecasting, and CFD provides an important tool in their production: from lava flow inundation (e.g., [1]) to tsunami inundation (e.g., [2]), from river overflow flooding (e.g., [3]) to sea ice dynamics (e.g., [4]), from landslides (e.g., [5]) to lahars (e.g., [6]); the range of phenomena that can be modeled is vast; and the tools can be employed both \"offline\", to determine preventative measures or to plan for short-term phenomena (e.g., [7]), as well as \"online\", to adapt intervention to the evolution of the phenomenon during a long-term event (e.g., [8]). A landslide is a mass movement that is generally triggered by intense rainfall, an earthquake, or slope cutting. One of the most hazardous types of landslide is a rapidly moving landslide (e.g., debris flow, debris avalanche, mudflow, flow slide and rock avalanche) due to its very short duration with potentially long run-out distance and massive damage to infrastructure and the population. In recent years, great advances have been made in the simulation of rapidly moving landslides using numerical models in order to forecast the flow direction, run-out distance, and velocity and to map the expected depths and area [9, 10]. Numerical models for fast-moving landslides are based on various assumptions and approaches. For example, the so-called depth-integrated models assume that a landslide behaves as a liquid mixture of interacting fluids and solids (e.g., [11, 12, 13, 14, 15, 16, 17, 18, 19]). Other numerical models use the Navier-Stokes equation for motion equations [20], while othersare based on the Savage-Hutter equation with different rheologies [16] or on smoothed particle hydrodynamics (SPH) to include rheologies describing the basal friction of Bingham, frictional, Voellmy, and cohesive-frictional viscous models [18]. Recently, a quasi-three-dimensional model was developed [21] that uses the Navier-Stokes equations and assumes that the sliding mass is an incompressible Newtonian fluid. In general, numerical modeling is strictly dependent on the knowledge of the rheological characteristics of the debris flow (e.g., the concentration of solid material, evolution of the mass movement, mixture density, viscosity, etc. ), as well as on the properties of the soil [22; 23; 24; 25]. Numerical modeling of lava flows has always been of great interest both for researchers contributing to the study of lava emplacement dynamics, as well as for stakeholders involved in environmental planning and the management of volcanic emergencies. Due to the high complexity of lava flow emplacement processes, numerical models necessarily adopt different simplifications and assumptions of the governing physical equations. Two kinds of lava flow models can be identified, mainly depending on the approach used, i.e., probabilistic/stochastic or deterministic. Probabilistic models provide information about the areas most likely affected by a lava flow based on multiple sets of input parameters, neglecting the temporal evolution of the flow and constraints on the thickness (e.g., [1; 26; 27; 28]). Deterministic models can instead simulate the spatiotemporal evolution of the flow, giving detailed information about the lava front advancement velocity and the final emplacement. The aforementioned models are all two-dimensional. However, 1D and 3D models have also been developed.