# Benchmarking Numerical Methods for Impact and Cratering Applications
## 4 Results and Discussion
### Benchmarking against Other Hydrocodes  
In this section, the FLAG and HOSS results are compared to those reported in the benchmarking study by [PERSON] et al. [5]. The study used multiple hydrodynamic/shock codes: ALE3D, AUTODYN, CTH, iSALE, RAGE, SOVA, SPH, and ZEUS-MP. These codes span a variety of approaches, including Eulerian, ALE, and smooth-particle hydrodynamics [5]. ALE3D is a hybrid (finite-element + finite-volume) multiphysics code developed at Lawrence Livermore National Laboratory and is based on ALE solutions. ALE3D has been used to simulate asteroid impacts, atmospheric entry, and breakup [51]. AUTODYN is part of the ANSYS Workbench package that was specifically designed to resolve the material response under high strain-rate loading conditions. For solid modeling, AUTODYN uses Lagrangian elements, while for fluid modeling, there is an option of using Eulerian formulations or smooth-particle hydrodynamics solvers. Researchers have used AUTODYN to simulate hypervelocity impacts of asteroids on the surface of the Earth [52]. CTH is a solid mechanics code that was developed at Sandia National Laboratories with a special focus on strong shock applications. CTH has been used to simulate many types of impact problems, such as those conducted for the Federal Emergency Management Agency [53] and for the National Aeronautics and Space Administration [54]. iSALE is an open-source hydrocode widely used by the planetary science community. It was developed for modeling fluid flows across a variety of speeds. However, it is largely serial and primarily used in 2D simulations [55; 56]. The RAGE hydrodynamics code, developed and maintained by Los Alamos National Laboratory, is an Eulerian hydrocode with capabilities for radiation hydrodynamics and AMR [57]. SOVA uses an Eulerian approach and models multi-material problems and dusty flows [58]. SPH has no underlying grid and can model solid materials [59]. Finally, ZEUS-MP is an Eulerian code that can model radiative and magnetized materials [60]. These codes are discussed in more detail in the benchmarking paper by [PERSON] et al. [5], but clearly this collection of shock codes represents a wide range of model formulations that can be used for impact cratering studies. Table 3 first compares the results from FLAG and HOSS to the theoretical maximum pressures for normal 5 km/s and 20 km/s impact velocities for the most resolved simulations. Note that the theoretical maximum pressure is derived from impedance matching in 1D for strengthless materials [3]. This table also includes the mean value, determined from all codes tested by [PERSON] et al. and their corresponding relative errors for comparison [5]. Note that the theoretical value applies only to the maximum pressure and not the shock pressure decay throughout the target. In addition, because the original comparison of [PERSON] et al. was published more than a decade ago, it is also expected that FLAG and HOSS would both show improvements from advances in numerical methods, model development, etc., that were not present in the other codes during the original study but may have since been implemented. Overall, the [PERSON] et al. results show relative errors of \(\sim\)25-30% [5]. Indeed, the FLAG and HOSS results both show improvement on these values, particularly in the 5 km/s case. In the 20 km/s case, the relative errors for FLAG and HOSS are somewhat larger than for the simulations with a 5 km/s impact velocity, consistent with the codes tested in the original study. \begin{table}
\begin{tabular}{c c c c c c} \hline  & & **Normal—5 km/s** & **45\({}^{\circ}\)—5 km/s** & **Normal—20 km/s** & **45\({}^{\circ}\)—20 km/s** \\ \hline \multirow{3}{*}{HOSS} & Number of cores & 1961 & 1961 & 1961 & 1961 \\  & Run time (hr:min) & 32:00 & 32:00 & 32:00 & 32:00 \\  & Simulated time (s) & 1.07 & 1.07 & 1.07 & 1.07 \\  & Number of cores & 360 & 360 & 360 & 360 \\ FLAG & Run time (hr:min) & 02:27 & 03:34 & 01:48 & 01:58 \\  & Simulated time (s) & 3.00 & 1.10 & 2.06 & 0.29 \\ \hline \end{tabular}
\end{table}