Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations

With numerous applications, particularly in fluid dynamics, the semi-Lagrangian approximation scheme is an essential part of the numerical analyst's toolkit. This largely self-contained book provides a framework for the semi-Lagrangian strategy for approximation of hyperbolic PDEs, with a special focus on Hamilton?Jacobi equations. The authors provide a rigorous discussion of the theory of viscosity solutions and the concepts underlying the construction and analysis of difference schemes; they then proceed to cover high-order semi-Lagrangian schemes and their applications to problems in fluid dynamics, front propagation, optimal control, and image processing. The text brings together developments from a wide range of sources to provide a unified treatment of the subject. This book is written for graduate and advanced undergraduate courses on numerical methods, and for researchers and practitioners whose work involves numerical analysis of hyperbolic PDEs.

Preface; Notation; 1. Models and motivations; 2. Viscosity solutions of first-order PDEs; 3. Elementary building blocks; 4. Convergence theory; 5. First-order approximation schemes; 6. High-order SL approximation schemes; 7. Fluid dynamics; 8. Control and games; 9. Front propagation; Bibliography; Index.

Maurizio Falcone is a Professor of Numerical Analysis in the Mathematics Department of Sapienza University of Rome. He is an associate editor for the journal Dynamic Games and Applications and was a member of the scientific board of the CASPUR Consortium for Scientific Computing (2002–2012) and on the steering committee of the ESF Network 'Optimization with PDE Constraints' (2008–2012). He is the author of about 60 papers in international journals. His main research areas are numerical analysis, PDEs, control theory and differential games, and image processing.
Roberto Ferretti is an Associate Professor in Numerical Analysis at Roma Tre University. He is the author of about 35 research papers in international journals and proceedings, mostly on semi-Lagrangian schemes. His main research areas are numerical analysis, PDEs, control theory, image processing, and environmental fluid dynamics.