Shock capturing and numerical dissipation in high-order methods for hyperbolic conservation laws

本项目旨在构建和研究双曲平衡律高阶数值方法中新的冲击捕获策略。在这种情况下，冲击捕获具有双重含义；介绍了对冲击的检测以及对其（鲁棒）处理的数值方案。冲击检测将基于不仅确定问题细胞而且确定跳跃不连续点精确位置的策略。将以Gelb和Tadmor的集中方法的新扩展，以及最近的多项式湮灭思想和现有策略的新组合为基础。由此获得的有关激波不连续点的位置和强度的信息将用于构建新的粘度项类别，与所有现有的粘度项相比，这些粘度项考虑了元件中激波的精确位置。这是第一次，数值耗散将只在确切需要的地方添加。最后，对新的黏度项及其离散性进行了分析和数值研究，并与已有的黏度项进行了比较。在这个项目结束时，我们希望提供新的，以前未知的冲击捕获策略。

This project is targeted at the construction and investigation of novelshock-capturing strategies in high-order numerical methods for hyperbolic balance laws. In this context shock-capturing has a dual meaning; referring to the detection of shocks as well as their (robust) treatment by the numerical scheme.Shock detection will be based on strategies which not only determine the troubled cells but also the precise location of the jump discontinuities. New extension of the concentration method of Gelb and Tadmor will be used as a basis, as well as the recent idea of polynomial annihilation and novel combinations of existing strategies.Thereby obtained information regarding the location and strength of a shock discontinuity will then be used to construct novel classes of viscosity terms which - in contrast to all existing ones - take into account the precise location of a shock in an element. For the first time, numerical dissipation will only be added where it is exactly needed. Finally, the new viscosity terms and theirdiscretisations will not only be investigated analytically as well as numerically but also compared with existing ones.At the end of this project, we expect to offer new, previously unknown shock-capturing strategies.