Modeling physics in computers requires discretizing continuous quantities in space. Approximations are key to simplify the abundance of dimensions in systems with millions of particles, waves, and their interactions across scales. One such approximation is to assume that a magnetized plasma behaves like a fluid. Theoretical physicists then assign densities, electromagnetic fields, and velocities to each point in a grid and ask one question over and over again: During a tiny step in time, how do balance laws – such as energy and momentum conservation – drive the change of these quantities at the interface between two cells in the numerical mesh? This question is called a Riemann problem and the information needed to answer it is encoded in partial differential equations, effectively the building block of all continuous dynamics. They dictate the speeds at which signals move within the fluid. The knowledge about these speeds allows us to answer the above question at each point in space very rapidly. A scientific simulation in many cases is a way to break down a complex problem into simple questions that can be asked over and over again – designed to model the correct physical reply in the limit of small enough grid cells and time steps.
Research summary
- We wrote a high-order numerical simulation tool for general relativistic force-free electrodynamics (the low inertia limit of relativistic ideal MHD) on the infrastructure of the Einstein Toolkit, making use of the fast parallelization infrastructure provided by the Carpet driver. See popularized animations below to get a feeling for why we need parallelization in astrophysical codes.
- Our code is highly accurate for the evolution of force-free plasma waves, so-called Alfvén and fast waves, and has been tested against a wide range of astrophysically relevant standard scenarios: Aligned and tilted dipole magnetospheres, aligned pulsar magnetospheres, black hole monopole and Wald magnetospheres.
- Monotonicity preserving (MP) discretization operators allow us to achieve very high order for the modeling of smooth force-free dynamics. We conducted an extensive review of the caveats of force-free electrodynamics – constrained to ideal electric fields and magnetic dominance – in current sheets, as to say in physically resistive layers.
- We provide implementations of both Cartesian and spherical meshes. The latter are well suited for curved geometries, as we find often in the magnetospheres of compact objects.
Visualizing science
In the earlier days of computer simulations, 1D and 2D setups dominated the field. They are much less demanding than their 3D counterpart, and the consecutive solution of Riemann problems could be done on the computational infrastructures available at the time. Even today, the largest simulations, such as the modeling of plasma turbulence with well above one billion grid cells can only be conducted in 2D. Still, this omits some of the dynamics emerging in the very important third dimension. However, it allows physicists to resolve both larger and very small structures at the same time.
In many situations, capturing the full dynamics of a physical problem requires all spatial dimensions. Like in the study of plasma waves and extended reconnection structures, where magnetic field lines wind up around each other and rearrange their topology. However, the amount of Riemann problems required to solve during a 3D simulation is much greater than in 1D and 2D. Large, high resolution simulations in 3D make use of a variety of techniques to become efficient for today’s computing infrastructures.
The most basic strategy is to decompose the 3D domain into many smaller subdomains. With the availability of large computers with thousands of computing units, even large 3D domains can be split up into many small simulations that can then solve all Riemann problems simultaneously. While this step is essential for all modern astrophysical simulations, it encompasses a great deal of progress in computer sciences, as all the computing units with small chunks of the simulation have to communicate to each other. The development towards more and more efficient decompositions and processing units is a very ‘hot topic’ and will enable even more powerful virtual laboratories in the future.
Collaborative results
J. F. Mahlmann, M. A. Aloy. V. Mewes, P. Cerdá-Durán, Computational general relativistic force-free electrodynamics: I. Multi-coordinate implementation and testing, Astronomy and Astrophysics, Volume 647, A57, March 2021, https://doi.org/10.1051/0004-6361/202038907
J. F. Mahlmann, M. A. Aloy. V. Mewes, P. Cerdá-Durán, Computational general relativistic force-free electrodynamics: II. Characterization of numerical diffusivity, Astronomy and Astrophysics, Volume 647, A58, March 2021, https://doi.org/10.1051/0004-6361/202038908
Jens Mahlmann 