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.
