Skip to main content

An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws

Periodic Reporting for period 2 - Extreme (An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws)

Reporting period: 2018-10-01 to 2020-03-31

"Nature is non-linear. Fundamental principles such as conservation of mass, momentum, and energy are mathematically modelled by non-linear time dependent partial differential equations. These advection-diffusion equations describe a wide range of problems in science and engineering, e.g. the prediction of noise and drag of modern
aircrafts, important for future mobility; the behaviour of gas clouds, important for the understanding of the interstellar
medium and star formation; the build up and propagation of tsunamis in oceans, important for prediction
and warning systems; the impact of the solar wind on Earth, important for communication satellites and
predictions of instabilities in plasma, important for future energy sources like fusion. In many cases, experimental studies of such problems are too costly, too time consuming, too dangerous, and are sometimes even impossible to perform. However, as knowledge of solutions are of utmost importance for modern
industry, science, medicine, and society, the numerical simulation of these problems has emerged as a key technology.
Due to their inherent low dispersion and dissipation errors, adaptive high order numerical methods currently receive growing attention in academia and industry. The potential benefits of this technologies are massively improved computational efficiency and drastic reduction in memory consumption that leads to both, (i) faster turn over times for existing simulation projects, and (ii), the enabling of novel large scale simulation projects that are currently not feasible. An indispensable property for the successful industrialisation of adaptive high order methods is robustness. Robustness, i.e. a computational solution framework that gives approximations for all reasonable simulation setups without crashing while retaining all the positive benefits of the high order methodology such as low numerical dissipation is the holy grail of the current research on these methods. Hence, the first challenge addresses the current state-of-the-art of robust high order accurate methods: is it possible to construct highly accurate methods that are robust, i.e. un-crashable, with minimal artificial dissipation? A mathematical path to achieve non-linear stability and robustness for discretisations of advection dominated problems is the concept of mathematical entropies and entropy solutions. Such solutions follow the second law of thermodynamics, namely that in every natural thermodynamic process the sum of all entropies increases. Our methological key to achieve such robustness is not intuitive at first sight: skew-symmetry. We will show that specific skew-symmetric formulations of the advection-diffusion problems, guided by careful mathematics, will allow us to design high order methods that are consistent with the second law of thermodynamics. Even with these new state-of-the-art adaptive high order methods, the range of scales in realistic applications
is enormous. Groundbreaking results can only be achieved with the power of the largest supercomputers. It is clear that the
algorithmic efficiency of a method is as important as its theoretical ""on-paper"" efficiency. Fortunately, the class of
high order methods offers a significant advantage compared to low order methods on modern computers, as their
algorithms consist of very dense operations with only little memory access. However, due to high algorithmic complexity, efficient high performance implementations are non-trivial and demand substantial re-design and re-engineering of the simulation software. Hence, the second challenge that we will address in this project is to design novel, exascale aware, adaptive algorithms and implement them in an open source solver that will scale on over 10^5 - 10^6 computing cores.
All of these developments will be performed in close collaborations with scientists from other research areas, such as e.g. engineering and natural sciences such as fluid mechanics, geophysics and astrophysics to get a direct feedback on the requirements of a state-of-the art pushing simulation framework."
The first main objective performed is the design, construction and analysis of a novel high order method, that is provably stable for the visco-resistive magnetohydrodynamics equations (VRMHD) on curvilinear unstructured hexahedral meshes. The VRMHD are among the most general non-linear advection-diffusion problems considered in this project. E.g. the famous compressible Navier-Stokes equations (CNS) are included as a special case and our methods are now also provably stable for the CNS. As an additional work performed, in collaboration with researchers from Max-Planck Institute for Plasma Physics, this novel method was implemented in an open source software, FLUXO, and is e.g. applied to investigate plasma instabilities occurring in setups important for fusion reactors. Another work performed is the application and extension of the computational framework to the simulation of Tsunami prediction. Here we also designed the new software such, that it can access the power of modern GPUs. These theoretical results were (and currently still are) extended to even more general methodologies, such as e.g. an implicit space-time formulation for the CNS that is provably entropy stable, a moving mesh framework that is provably entropy stable and non-conforming approximation spaces needed for adaptivity that are provably entropy stable.
The development of skew symmetric entropy stable high order variants pushed the state-of-the-art in the development of discontinuous Galerkin (DG) methods and allowed to consider additional structure preserving properties for the first time, such as e.g. kinetic energy preservation (KEP) and energy compatibility. In collaboration with colleagues from engineering (University of Stuttgart), it could be demonstrated that a novel large eddy simulation DG approach based on KEP and explicit turbulence modelling is superior to the current state-of-the-art implicit turbulence modelling approach for DG. We could demonstrate that the novel DG schemes are indeed more robust than what was available before, in collaboration with researchers from Imperial College. It was further possible to construct the first provably entropy stable stable fully discrete high order scheme that does not rely on the assumption of exact numerical integration. It could be demonstrated in case of the two-dimensional shallow water equations used to predict the arrival times of Tsunami waves that the new algorithms are extremely well suited to modern many-core architectures such as GPUs. We could demonstrated that the novel algorithms are so cache effective, that the overhead in additional number of operations from the skew-symmetric forms is completely redeemed by modern GPUs with very high sustained peak performances. Due to these high potential and demonstrated gains, the novel high order DG and related methods are on the rise in the DG community right now and many related research activities emerge. Besides the more theoretical developments in the project, the second phase of the project will strongly focus on the extension of the open source simulation frameworks and on stronger interaction with potential collaborators from other research areas such as engineering and natural sciences. It is expected to see novel large scale simulations with state-of-the-art pushing fidelity.