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

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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.