Periodic Reporting for period 4 - Extreme (An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws)
Reporting period: 2021-10-01 to 2022-03-31
Non-linear 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. For such problems, adaptive high order discretizations offer great potential benefits, such as massively improved computational efficiency and drastic reduction in memory consumption. 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. Hence, the main theme of this project is: is it possible to construct highly accurate methods that are robust and nigh uncrashable? The general objective is to construct high order schemes that have additional provable mathematical properties such as entropy consistency and energy consistency. Furthermore, mechanisms to handle large solution gradients (shocks) and that guarantee physical bounds of solutions (e.g. positivity of density and pressure) are necessary. Last but not least, these novel methods should give algorithms that are very well suited for high performance computing, to enable the largest super computers and allow for large scale simulations on clusters. Concluding the actions of this project, basically all of the aboce mentioned objectives could be achieved, with novel schemes and algorithms. Furthermore, all of these mathematical developments are implemented in highly efficient open source software and applied to many challenging examples such as, e.g. simulation of the propagation of a Tsunami, interaction of Jupiter moon Io with its space environment, and a supernovae explosion.
Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far
The work was split in mainly two parts: (i) theoretical development, construction, and analysis of novel simulation algorithms; (ii) implementation, validation, testing and application of the novel algorithms in open source software. Many small steps in each of these categories have been performed throughout the project, all fitting together to reach the global objectives. An entropy stable adaptive high order scheme for the visco-resistive magnetohydrodynamics equations has been developed. These equations can be used to model (space) plasma. These methodology was extended to 3D curvilinear grids to allow for complex geometry in the application. Furthermore, these methods are constructed such, that inner, magnetic and kinetic energies are consistently preserved and compatible to the total energy. Furthermore, entropy consistent shock capturing has been derived, that allows for a local blending of a compatible provably monotone low order method with the high order method. This allows to handle strong shock waves and furthermore enabled to include a mechanism that preserves positivity of important physical quantities, such as density and pressure. Hence, a combination of all these technologies make for an extremely robust, but accurate simulation framework and the goal is almost met. The only caveat is that mathematically, a very strict explicit time step restrictions is necessary to guarantee these robustness properties. To boost overall performance of the methodology adaptivity was included and made compatible with the entropy estimates. To extend impact and outreach of this more methodology focused research, a new simulation software in the very accessible and novel programming language Julia has been implemented. This software project is already quite successful with a growing national and international developer and user base. To increase the scientific impact even more, variants of the novel methods have been implemented in the astrophysics community code FLASH, where now an open source module with a discontinuous Galerkin solver is now available for the scientists to experiment with. Finally, these methods have been applied to challenging 'lighthouse' applications from natural sciences including complex multi-physics simulations. First, simulation of the Jupiter moon Io in its plasma space environment has bee investigated. Here, the idea is to compare to magnetic field measurements of spacecrafts, which try to detect if there are large reservoirs of water under the surface. The second application was in the area of the simulation of interstellar medium. Here, variants of the novel methodolgy have been connected to a fully multi-physics framework that includes self-gravity, raditation transfer, and multiple species with chemical networks. All of these software and application developments have be performed in close collaborations with the domain scientists to get a direct feedback on the requirements of a novel simulation framework.
Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)
The development of entropy stable high order methods on curvilinear grids in 3D for visco-resistive MHD. Novel skew-symmetric split formulations have been developed for discontinuous Galerkin methods that are kinetic energy preserving. It turned out that these methods are highly robust for compressible turbulent flows and that these methods are in particular well suited for explicit modelling of subgrid scale turbulence. Furthermore, a provably entropy stable convex blending of high order LGL collocation discontinuous Galerkin methods with a compatible LGL subcell finite volume approach was developed,that allows for shock capturing and positivity preservation. The novel simulation software project Trixi.jl was the first available tool written in the novel programming language Julia that demonstrates that Julia can be as fast (or faster) for complex 3D simulations of non-linear partial differential equations with high order DG methods compared to well established simulation tools based on Fortran or C. The novel FLASH module with a DG solver is the first fully compatible multi-physics implementation within the astrophysics community.