Analysis and Numerics for Conservation Laws

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What do a supernova explosion in outer space, flow around an airfoil, and knocking in combustion engines have in common? Despite their different physical and chemical mechanisms, scales, and motivations for study, they share a common thread: the underlying fluid flows can be described by similar hyperbolic systems of partial differential equations known as conservation laws. Supernovae, which are thermo-nuclear explosions at a scale of 10 cm, are studied by astrophysicists to gain insights into fundamental properties of the universe. In contrast, shock waves around airfoils of commercial airliners, occurring at a scale of 3 cm, affect wing stability and fuel consumption, necessitating precise engineering designs. Knocking in combustion engines, a chemical process at a scale of 10 cm, must be minimized to prevent engine damage, optimizing for efficiency and environmental considerations. Astrophysicists, engineers, and mathematicians are united in their pursuit of advancements in theory related to these equations and the development of computational methods for their solutions. Partial differential equations are a major research area in mathematics, with a significant focus on analyzing and numerically approximating solutions. Hyperbolic conservation laws in multiple dimensions continue to present one of the main challenges in modern mathematics.