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

本计画的目标是在高阶数值方法中建构和研究双曲平衡定律的新颖激波捕捉策略。在这种情况下，激波捕捉具有双重意义，即激波的检测和数值格式对激波的（鲁棒）处理。激波检测将基于不仅确定故障单元而且确定跳跃不连续性精确位置的策略。Gelb和Tadmor浓度方法的新扩展将用作基础，以及最近提出的多项式湮灭的概念和现有策略的新组合。由此获得的关于激波不连续性的位置和强度的信息将用于构造新的粘性项类别，与所有现有的粘性项相比，要考虑到冲击波在元件中的精确位置。这是第一次，数值耗散将只添加在真正需要的地方。最后，新的粘性项和它们的离散化不仅将被解析地和数值地研究，而且还将与现有的粘性项和它们的离散化进行比较。在这个项目的最后，我们期望提供新的、以前未知的激波捕获策略。

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.