AEI–DFG: Structure preserving genuinely multi-dimensional high order finite-volume methods applied to balance laws with non-conservative terms

• Applicant: Professor Christian Klingenberg, Ph.D.
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541675783

The main objective of this project is the development of genuinely multidimensional numerical schemes of finite volume type for solving hyperbolic systems, which are able to produce accurate results on coarse grids. By genuinely multidimensional we mean taking into account multidimensional features of the flows, which make them much more complex and difficult than their one-dimensional counterparts. An example are vortices in compressible flow. To do that, two different strategies will be followed. The first one consists in introducing multidimensional information in the numerical fluxes through the approximate solution of multidimensional Riemann problems at corners of the computational cells. The second approach is based in the so-called Active Flux method, in which not only cell averages are updated, but also additional degrees of freedom consisting of point values on the cell boundary. High-order versions of both families of schemes will be analyzed and systematically compared, for systems of conservation laws and balance laws with source and coupling terms. To unify our study, the schemes will be formulated in the more general framework of nonconservative hyperbolic systems. Moreover, the construction of well-balanced schemes, able to exactly preserve all or some specific families of stationary solutions, will also be addressed. In order to develop schemes that are able to provide accurate results in different flow regimes, we will investigate implicit or semi-implicit versions of the proposed multidimensional methods. Finally, applications to some interesting systems will be considered, e.g., Euler equations with gravity, magnetohydrodynamics, shallow water systems with Coriolis force, and multilayer shallow water systems with bottom topography.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partners: Professor Dr. José-María Gallardo-Molina; Professor Dr. Tomás Morales de Luna