Introduction

Rudimentary elements of the Pencil Code were originally developed at the Helmholtz Institute for Supercomputational Physics in Golm, Germany. It began during the Summer School “Tools to Simulate Turbulence on Supercomputers” held 27 August – 21 September 2001 within the premises of the Albert Einstein Institute in Golm; see its annual report (Nicolai 2001). The spatial and temporal discretisation schemes used in the code are described by Brandenburg and Dobler (2002) and Brandenburg (2003), which have become the most commonly quoted sources with reference to the Pencil Code. In the meantime, however, more than seventy people have contributed to the further development of the code,¹ such that some revised and more comprehensive references are badly needed. It is towards this end that we present this special issue in Geophysical and Astrophysical Fluid Dynamics (GAFD) to discuss specific applications and their numerical aspects, especially relating to newly emerging research topics.

Early applications of the Pencil Code were in dynamo theory of forced turbulence in Cartesian domains (Haugen et al. 2003, 2004a, 2004b). Subsequently, continued code developments have extended its scope in many different directions. On one hand, cylindrical and spherical geometries have been employed in applications ranging from mean-field and forced turbulence simulations (Mitra et al. 2009, 2010, Kemel et al. 2011) to convectively driven dynamos (Käpylä et al. 2010, 2012, 2013) and circumstellar disks (Lyra and Mac Low 2012, Lyra et al. 2015, 2016). On the other hand, Cartesian geometries have been used in more complex physical settings ranging from technical applications to a broad spectrum of astrophysical applications. This special issue of GAFD gives insight into some of the problems with which members of the Pencil Code community are currently concerned. Some of the topics relate to the necessary physical setup, in order to address specific questions in the most appropriate manner, while others concern the solutions to numerical challenges that have been encountered in the process of solving various problems.

Specifically, we begin the special issue by addressing the global convection-driven dynamo problem in spherical geometry. The actual implementation of spherical geometry has been described in an earlier paper by Mitra et al. (2009), which involves replacing partial derivatives by covariant derivatives. As an example, the traceless rate-of-strain tensor is generalised by substituting (1) $\frac{1}{2}(u_{i,j}+u_{j,i})-\frac{1}{3}{\delta }_{ij}{u}_{k,k}\to \frac{1}{2}(u_{i;j}+u_{j;i})-\frac{1}{3}{\delta }_{ij}{u}_{k;k},$(1) where commas and semicolons denote partial and covariant derivatives, respectively. The physical challenge of addressing global spherical dynamo problems concerns the vast separation of time scales from minutes in the surface granulation to years in deep layers governed by the geometric mean of dynamical and Kelvin-Helmholtz time scales (Spiegel 1987). This problem is approached by modifying the physical setup such that a model star has time scales that are compressed to a much narrower range. This can lead to artifacts that need to be studied in detail. Addressing this question is the topic of the paper by Käpylä et al. (2020).

Among the technical applications of the Pencil Code are the presence of solid objects within the fluid flow, such as, e.g. a cylinder in a cross flow, and the use of more complex boundary conditions. In this connection, Aarnes et al. (2020) have compared two different approaches for handling solid bodies within a fluid flow. One is the immersed boundary method and the other is the overset grid method. Both are implemented in the code and have been compared with regard to performance and accuracy. The overset grid method is found to be superior to the immersed boundary method and has now been used to study particle impaction on a cylinder in a cross flow (Aarnes et al. 2019).

Another technical application concerns hydrogen–oxygen combustion in rockets, for example, but this can sometimes also lead to detonation. Simulating this is a challenging problem, whose success depends crucially on being able to resolve the pressure and chemical reaction fronts well enough so that they do not separate and detach from each other, which can quench the detonation. This has been addressed in the paper by Qian et al. (2020). Here, solutions with the Pencil Code have been resolved with mesh widths down to 0.2 micrometers in a one-dimensional domain of 10 cm length. This corresponds to a computational work load of half a million mesh points and several days of wall clock time on 2048 processors.

The numerical handling of shocks with artificial viscosity is a main topic of the paper by Gent et al. (2020). In addition to presenting several one-dimensional Riemann shock tube problems, the authors also show the detailed analysis of the evolution of individual three-dimensional supernova remnants. These results are relevant to the numerical treatment of interstellar turbulence driven by supernova explosions, in which thousands of such events are required over millions of years and longer. Spatial and temporal resolution cannot therefore be unlimited, and the study assesses the physical veracity of coarse grain models practicable for inclusion in such turbulence simulations.

On cosmological scales, Schober et al. (2020) focus on relativistic plasmas in the early universe. A relativistic plasma is one where the chirality of fermions can play a dominant role and produce additional effects analogous to the α effect in mean-field electrodynamics (Moffatt 1978, Krause and Rädler 1980), but now at the fully microphysical level. This leads to modifications of the governing magnetohydrodynamic (MHD) equations that have been solved with the Pencil Code during the last few years (Brandenburg et al. 2017, Rogachevskii et al. 2017, Schober et al. 2018). In this issue, Schober and collaborators invoke yet another effect, called the chiral separation effect, and describe its implementation and tests.

Again on cosmological scales, Roper Pol et al. (2020) discuss for the first time the direct numerical solution of gravitational waves from hydrodynamic and MHD turbulence driven during the electroweak phase transition, when the Universe was just some ${10}^{-11}\mathrm{s}$ old. The authors identify an important problem that concerns the numerical accuracy of the gravitational wave spectrum at large wavenumbers. A brute force approach would require time steps that are about 20 times shorter than what is expected based on the usual Courant-Friedrich-Levy condition. To avoid this serious restriction, they solve the gravitational wave equation analytically in Fourier space between two subsequent time steps. Their approach leads to a speed-up by about a factor of ten.

Another problem related to the time step is tackled in the paper by Brandenburg and Das (2020), where the authors solve the radiation hydrodynamics equations with the Pencil Code for hot stars, in which the radiation pressure becomes important, and for accretion disks around white dwarfs. Their paper gives a detailed characterisation of the empirical time step constraints in a broad range of different circumstances and also comparisons with theory.

Returning to the Sun, several applications with the Pencil Code have previously determined the oscillation frequencies in stratified layers (Singh et al. 2014, 2015). In their new work, Singh et al. (2020) present results for a more complicated magnetic field geometry. In particular, they find that the unstable Bloch modes reported previously (Singh et al. 2014) are now absent when more realistic localised flux concentrations are considered.

Chatterjee (2020) considers realistic solar setups of the solar atmosphere and solves for the generation, propagation, and dissipation of Alfvén waves in the solar atmosphere. The displacement current is retained to limit the propagation speed and thus the time step. This is done by using the Boris correction (Boris 1970), which treats the displacement current in a semi-relativistic manner and was implemented in the Pencil Code by Chatterjee.

Bourdin (2020) considers realistic coronal setups and discusses the use of a novel boundary condition to treat non-vertical and non-potential magnetic fields. He also discusses the scalability of the code, with the introduction of the extendable HDF5 file format and input-output strategies that are better suited for modern supercomputers.

Warnecke and Bingert (2020) further improved such a model of the solar corona by using the Boris correction implemented by Chatterjee (2020) and further characterise its properties in their work on non-Fickian or non-Fourier heat conduction using the telegraph equation. This approach is analogous to what was done previously to solve for cosmic ray diffusion in the interstellar medium (Snodin et al. 2006) with a very large diffusion coefficient, which would severely limit the time step. As in earlier work (Blackman and Field 2003, Brandenburg et al. 2004, Rempel 2017), the authors also found a significantly increased time step, which resulted in a speed-up of the code.

As already alluded to in the opening paragraphs, this special issue gives only a glimpse at some of the applications of the Pencil Code to a diverse range of problems. There are many other papers, where further aspects of the code are presented. As an example, we mention the work of Johansen et al. (2007, 2011, 2012) and Lyra et al. (2008, 2009, 2015, 2016) on planet formation and papers by Li et al. (2017, 2018, 2019) on raindrop formation. These papers discuss the computation of inertial particles in a turbulent flow with applications to astrophysics and meteorology. A complete list of the currently 527 papers that refer to the Pencil Code was presented by Brandenburg (2019b) at the Pencil Code User Meeting 2019 in Espoo (Finland).

The code has also been used to compute turbulent front propagation, which results in a spatial generalisation to reaction-diffusion equations describing the emergence of homochirality (Brandenburg and Multamäki 2004). Interestingly, one tends to think of the Pencil Code as a tool for solving partial differential equations, but even this is not always the case and one may just solve ordinary differential equations (ODEs). An advantage of employing the Pencil Code technology lies in the ease at which many ODEs can be solved at the same time on many processors. This can become a significant advantage for solving many realizations of stochastic differential equations, as done in a recent alternative approach to solving astrobiological reaction equations to describe the evolution toward homochirality (Brandenburg 2019a).

Another major Pencil Code activity of the past decade concerns the test-field method. This has been reviewed on an earlier occasion (Brandenburg et al. 2010) and has also been applied to convection-driven turbulent dynamos in spherical geometry (Warnecke et al. 2018). Solving not just one set of test fields, but different ones in isolation or simultaneously is particularly straightforward with the Pencil Code owing to its modularity (Brandenburg et al. 2012). Similarly, exploring the parallels between direct numerical simulations and mean-field simulations has played a major role in the investigation of the negative effective magnetic pressure instability (Kemel et al. 2012, 2013). Here the exploration of both approaches is particularly useful for understanding relevant parameter regimes, for example.

In conclusion, we hope that this special issue does some justice to the community of Pencil Code users, and especially the developers who all work primarily toward their own research goals, but they do so in such a way that their efforts benefit the other users in a constructive manner. This is made feasible through sets of automated test runs that verify the integrity of the code hourly, daily, and even after every single update to the code. At the time of writing this editorial, 30,431 updates or commits have been made by the code developers. In fact, 34 developers have done more that 34 commits, which is like an h index of the Pencil Code. This underlines the special role of this code development as a community effort in sharing and utilising each others work.

Notes

1 https://github.com/orgs/pencil-code/people