Advances in Numerical Methods for Hyperbolic Balance Laws and Related Problems

A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021.

Chapter 1. Alessandro Alla, Peter M. Dower, Vincent Liu. A Tree Structure Approach to Reachability Analysis.- Chapter 2. Giulia Bertaglia. Asymptotic-preserving neural networks for hyperbolic systems with diffusive scaling.- Chapter 3. Felisia Angela Chiarello, Paola Goatin. A non-local system modeling bi-directional traffic flows.- Chapter 4. Armando Coco, Santina Chiara Stissi. Semi-implicit finite-difference methods for compressible gas dynamics with curved boundaries: a ghost-point approach.- Chapter 5. Elena Gaburro, Simone Chiocchetti. High-order arbitrary-Lagrangian-Eulerian schemes on crazy moving Voronoi meshes.- Chapter 6. Elisa Iacomini. Overview on uncertainty quantification in traffic models via intrusive method.- Chapter 7. Liu Liu. A study of multiscale kinetic models with uncertainties.- Chapter 8. Fiammetta Conforto, Giorgio Martalò. On the shock wave discontinuities in Grad hierarchy for a binary mixture of inert gases.- Chapter 9. Giuseppe Visconti, Silvia Tozza, Matteo Semplice, Gabriella Puppo. A conservative a[1]posteriori time-limiting procedure in Quinpi schemes.- Chapter 10. Yuhua Zhu. Applications of Fokker Planck equations in machine learning algorithms.

Giacomo Albi is Associate Professor of Numerical Analysis at the Department of Computer Science, University of Verona. He received his Ph.D. from the University of Ferrara. He was recipient of the 2014 Copernico award and the UMI-INdAM-SIMAI 2017 prize. He worked at TU Munich on the project "High-Dimensional Sparse Optimal Control". His research focuses on numerical methods for kinetic equations, hyperbolic balance laws, and control of multi-agent systems.

Walter Boscheri is Associate Professor of Numerical Analysis at the University of Ferrara, Italy. His research is concerned with the development and implementation of numerical methods for partial differential equations on fixed and moving unstructured meshes. He designs novel high order finite volume and discontinuous Galerkin schemes with structure- and asymptotic-preserving properties applied to continuum mechanics, including implicit-explicit time discretizations.

Mattia Zanella is Associate Professor of Mathematical Physics at the Department of Mathematics "F. Casorati" of the University of Pavia. He was recipient of the Copernico award in 2018 and the Anile Prize in 2019. In 2019 he got a fellowship from the Hausdorff Research Institute for Mathematics. His research interests are focused on uncertainty quantification, optimal control and kinetic modelling of  collective phenomena with applications in physics and life science.