High-order accurate numerical methods for hyperbolic partial differential equations on Cartesian grids with embedded geometry

• Applicant: Professorin Dr. Christiane Helzel
• Subject Area: Mathematics
• Term: from 2016 to 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 325695158

• Final Report Year: 2025

Final Report Abstract

The project is devoted to the development of high order accurate finite volume methods for hyperbolic partial differential equations. The main goals are: (1) the development of new third order accurate finite volume methods, and (2) the construction of third order accurate Cartesian grid cut cell methods for the approximation of hyperbolic problems in complex geometries. In particular our work is concerned with the development of fully discrete Active Flux methods which have a compact stencil in space and time. This property is essential for the derivation of new and accurate Cartesian grid cut cell methods.

Publications

• A New ADER Method Inspired by the Active Flux Method. Journal of Scientific Computing, 80(3), 1463-1497.
  Helzel, Christiane; Kerkmann, David & Scandurra, Leonardo
  
• A Third-Order Accurate Wave Propagation Algorithm for Hyperbolic Partial Differential Equations. Communications on Applied Mathematics and Computation, 2(3), 403-427.
  Helzel, Christiane
  
• An Active Flux Method for Cut Cell Grids. Springer Proceedings in Mathematics & Statistics, 507-515. Springer International Publishing.
  Helzel, Christiane & Kerkmann, David
  
• Active Flux Methods for Conservation Laws on Complex Geometries, Dissertation Heinrich-Heine-Universität Düsseldorf
  D. Kerkmann
  
• The Cartesian Grid Active Flux Method: Linear stability and bound preserving limiting. Applied Mathematics and Computation, 393, 125501.
  Chudzik, Erik; Helzel, Christiane & Kerkmann, David
  
• A Review of Cartesian Grid Active Flux Methods for Hyperbolic Conservation Laws. Springer Proceedings in Mathematics & Statistics, 93-109. Springer Nature Switzerland.
  Chudzik, Erik & Helzel, Christiane
  
• The Cartesian Grid Active Flux Method with Adaptive Mesh Refinement. Journal of Scientific Computing, 94(3).
  Calhoun, Donna; Chudzik, Erik & Helzel, Christiane
  
• Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics. Journal of Scientific Computing, 99(1).
  Chudzik, Erik; Helzel, Christiane & Lukáčová-Medvid’ová, Mária
  
• Cartesian Grid Active Flux Methods for Hyperbolic Conservation Laws, Dissertation Heinrich-Heine-Universität Düsseldorf
  E. Chudzik
  
• A positivity-preserving Active Flux method for the Vlasov-Poisson system. Journal of Computational Physics, 524, 113693.
  Kiechle, Yanick; Chudzik, Erik & Helzel, Christiane