This volume gathers contributions reflecting topics presented during an INDAM workshop held in Rome in May 2016. The event brought together many prominent researchers in both Mathematical Analysis and Numerical Computing, the goal being to promote interdisciplinary collaborations. Accordingly, the following thematic areas were developed: 1. Lagrangian discretizations and wavefront tracking for synchronization models; 2. Astrophysics computations and post-Newtonian approximations; 3. Hyperbolic balance laws and corrugated isometric embeddings; 4. “Caseology” techniques for kinetic equations; 5. Tentative computations of compressible non-standard solutions; 6. Entropy dissipation, convergence rates and inverse design issues. Most of the articles are presented in a self-contained manner; some highlight new achievements, while others offer snapshots of the “state of the art” in certain fields. The book offers a unique resource, both for young researchers looking to quickly enter a given area of application, and for more experienced ones seeking comprehensive overviews and extensive bibliographic references. Laurent Gosse received the M.S. and Ph.D. degrees both in Mathematics from Universities of Lille 1 and Paris IX Dauphine in 1991 and 1997 respectively. Between 1997 and 1999 he was a TMR postdoc in IACM-FORTH (Heraklion, Crete) mostly working on well-balanced numerical schemes and a posteriori error estimates with Ch. Makridakis. From 1999 to 2001, he was postdoc in Universtity of L’Aquila (Italy) working on stability theory for systems of balance laws and multiphase computations in geometrical optics with K-multibranch solutions. In 2001, he moved to University of Pavia working on asymptotic-preserving schemes and degenerate parabolic equations with G. Toscani. In 2002, he was granted a permanent researcher position at CNR in Bari (Italy) where he developed Lagrangian schemes for nonlinear diffusion models and a stable inversion algorithm for Markov moment problem with O. Runborg. Numerical investigation of semiclassical WKB approximation for quantum models of crystals was conducted with P.A. Markowich between 2003 and 2006. Since 2011, he holds a CNR position at both Roma and University of L’Aquila and works mainly on the applications of Caseology to well-balanced schemes for collisional kinetic equations. Roberto Natalini received his PhD in Mathematics from the University of Bordeaux I (France) in 1986. He has been the Director of the National Research Council of Italy’s Istituto per le Applicazioni del Calcolo since 2014. His research interests include: analysis of hyperbolic systems of conservation laws, fluid dynamics, road traffic, semiconductors, chemical damage to monuments, and biomathematics. He serves on the Board of the Italian Society of Industrial and Applied Mathematics and is Chair of the European Mathematical Society’s Raising Awareness Committee. 1 A nonlocal version of wavefront tracking motivated by Kuramoto-Sakaguchi equation.- 2 High-order post-Newtonian contributions to gravitational self-force effects in black hole spacetimes.- 3 Concentration waves of chemotactic bacteria: the discrete velocity case.- 4 A numerical glimpse at some non-standard solutions to compressible Euler equations.- 5 On Hyperbolic Balance Laws and Applications.- 6 Viscous equations treated with L-splines and Steklov-Poincar operator in two dimensions.- 7 Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation.- 8 A well-balanced scheme for the Euler equations with gravitation.- 9 Practical convergence rates for degenerate parabolic equations.- 10 Analysis and simulation of nonlinear and nonlocal transport equations.- 11 Semi-analytical methods of solution for the BGK-Boltzmann equation describing sound wave propagation in binary gas mixtures.- 12 Convergent Lagrangian discretization for drift-diffusion with nonlocal aggregation
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