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Lecturer
Davide Modesti
Timetable and workload
Lectures: 24 hours
Apr 20- May 22, The timetable will be provided later on
Course content
This course provides an introduction to the numerical analysis of hyperbolic conservation laws. The focus is primarily on one-dimensional scalar conservation laws. Classical solutions are first introduced through the method of characteristics, followed by the concept of weak solutions and the associated jump (Rankine–Hugoniot) conditions. Issues of well-posedness are briefly discussed, with particular emphasis on uniqueness and the Oleinik entropy condition.
The study of numerical methods begins with the linear advection equation. Numerical schemes derived from the method of characteristics are introduced, and the fundamental concepts of stability, consistency, and convergence are discussed, leading to the Lax equivalence theorem. The course then addresses numerical methods for nonlinear conservation laws within the framework of finite difference schemes. Conservative formulations are emphasized, and the Lax–Wendroff theorem for conservative schemes is presented, together with flux-splitting methods.
The course concludes with the study of high-resolution schemes for hyperbolic problems, including Total Variation Diminishing (TVD) methods, Sweby-type schemes, and modern non-oscillatory approaches such as ENO and WENO schemes.
The lectures will be complemented by practical sessions where the students will implement the numerical schemes discussed during the course.
References
[1] Numerical Methods for Conservation Laws — Randall J. LeVeque
[2] Finite Volume Methods for Hyperbolic Problems — Randall J. LeVeque
[3] Riemann Solvers and Numerical Methods for Fluid Dynamics — Eleuterio Toro