Winter School

Winterschool part II - Advanced numerical methods for hyperbolic equations 2024

29 gennaio 2024
30 gennaio 2024
31 gennaio 2024
1 febbraio 2024
2 febbraio 2024
Orario di inizio 
Polo di Mesiano - Via Mesiano 77, Trento
Aula 2A and Online
Organizzato da: 
Prof. Michael Dumbser - Department of Civil, Environmental and Mechanical Engineering
A pagamento con prenotazione
Scadenza prenotazioni: 
25 gennaio 2024, 12:00
Michael Dumbser
Lorena Galante


Michael Dumbser and Laura Del Rio

Department of Civil and Environmental Engineering
Laboratory of Applied Mathematics
University of Trento, Italy

During the course, two special lectures of 1h each will be given by Prof. Dr. Dr. hc. Eleuterio F. Toro, OBE


The short course on advanced numerical methods consists of a structured intensive one-week program of 40 hours of theoretical lectures and computer laboratory exercises on advanced numerical methods for hyperbolic partial differential equations with applications in engineering and science. The course covers finite volume methods, the exact and approximate solution of the Riemann problem, second-order TVD methods, higher-order ENO, WENO, and discontinuous Galerkin methods, as well as the discretization of non-conservative problems. Special emphasis is put also on numerical methods that are able to handle complex geometries. In particular, unstructured Finite Volume and discontinuous Galerkin schemes as well as mesh-free particle methods are presented. The course is primarily designed for PhD students and post-doctoral researchers in applied mathematics, engineering, physics, computer science, and other scientific disciplines. The course may also be of interest to senior researchers in industry and research laboratories, as well as to senior academics. The lectures on the theory will be supplemented with laboratory-based computer exercises to provide hands-on experience to all participants on the practical aspects of numerical methods for hyperbolic problems and applications using MATLAB programs specially designed for the course.


Review of basic theoretical aspects of hyperbolic conservation laws and numerical concepts for hyperbolic equations. Finite volume methods for one-dimensional systems. Godunov's method. The Riemann problem. Approximate Riemann solvers. Godunov-type finite volume methods for non-linear systems. Construction of higher order non-oscillatory methods via non-linear schemes: TVD, ENO and WENO reconstruction procedures. Discontinuous Galerkin Finite Element methods for one-dimensional problems. The well-balanced property and numerical methods for non-conservative hyperbolic systems. Extension to multiple space dimensions on Cartesian grids. 

Complex geometries using unstructured triangular meshes in two space dimensions and using mesh-free approaches. 
Mesh-based algorithms: Finite volume schemes on unstructured meshes for two-dimensional geometries. Second-order reconstruction and slope limiting on unstructured meshes. Applications to the shallow water equations and the Euler equations of compressible gas dynamics. High-order discontinuous Galerkin finite element methods on unstructured meshes. 
Mesh-free algorithms: Introduction to Lagrangian particle methods. Guidelines for implementation of smooth particle hydrodynamics (SPH) based on approximate Riemann solvers.

On the last day, the course is rounded off by advanced seminar-style lectures with outlooks to the following topics: better than second-order schemes on unstructured meshes, high-order methods on space-time adaptive grids (AMR), time-accurate local time stepping (LTS), high order Lagrangian schemes on moving unstructured meshes, applications to compressible multi-phase flows and nonlinear elasto-plasticity. Numerical methods for all Mach number flows. 

About Trento and the Dolomites

The historical city of Trento is situated in the autonomous Italian region of Trentino - Südtirol, close to the world-famous mountains called Dolomites. Trento is very easy to reach by car or train from Austria (150 km south of Innsbruck) and from Verona (90 km north of Verona). The nearest and most convenient airport is Verona Airport, 15 minutes from the Verona train station. The region around Trento is of extraordinary beauty, with its unique mountains and lakes that offer the participants many exciting outdoor activities like skiing, hiking or climbing.


Onsite participation fee

  • Students and post-docs €500
  • Senior academics and others €1000
  • Participation is free for students and post-docs of UniTN

Fees cover lectures, laboratory exercises, lecturing material and MATLAB sample programs. 

Online participation fee

  • For all €250. 

The fee covers lectures, laboratory exercises, lecturing material, and MATLAB programs. Lectures and exercises are transmitted via ZOOM.

All the fees are free of "VAT tax” as art. 10 DPR 633/72.

IAHR members will receive a discount on the fee. Once you have registered for the course, you must send us an email to indicate that you are a member to apply for the discount.

Registration and administrative information

All participants (onsite and online) must register online.

All details concerning the payment of the course fee will be sent individually to each participant after registration. The information will be sent via e-mail after January 1, 2024

Payment of the course fee must be made only after registration and after January 1st, 2024, but before January 27th, 2024.

If payment of the registration fee will be made by an Italian Public Administration we invite you to fill out the Invoice data form that you can find in the download box.

For foreigners who wish to receive a formal letter of acceptance to the Winter School, they are invited to send an email to lorena.galante [at] with a specific request.

Onsite participants must bring their laptops with MATLAB installed. Online participants will receive the ZOOM link of the course only after payment of the course fee.


The course received support from the International Association for Hydro-Environment Engineering and Research (IAHR)

PDF icon TimeTable 2024 (PDF | 141 KB)