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

This largely self-contained book provides a unified framework of semi-Lagrangian strategy for the 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 high-order semi-Lagrangian schemes and their applications to problems in fluid dynamics, front propagation, optimal control, and image processing. The developments covered in the text and the references come from a wide range of literature.

Audience: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations is written for advanced undergraduate and graduate courses on numerical methods for PDEs and for researchers and practitioners whose work focuses on numerical analysis and numerical methods for nonlinear hyperbolic PDEs.

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