Short course on Advanced numerical methods for hyperbolic equations 2026

Lecturers

• Prof. Dr.-Ing. Michael Dumbser
• Prof. Olindo Zanotti
• Two special lectures will be given by Prof. Dr. Dr. hc. E.F. Toro, OBE Department of Civil and Environmental Engineering Laboratory of Applied Mathematics University of Trento, Italy 

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

Summary

The short course on advanced numerical methods consists of a structured, intensive one-week programme 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 discretisation of nonconservative problems. Special emphasis is also placed on numerical methods that can 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. 

Contents

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 elastoplasticity. 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 the Dolomites. Trento is easily accessible by car or train from Austria (approximately 150 km south of Innsbruck) and from Verona (approximately 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 participants many exciting outdoor activities such as skiing, hiking or climbing.

Fee

On-site participation fee:

• Students and post-docs €500
• Senior academics and others €1000

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

IAHR members will receive a 10% discount on the onsite participation fee (not online participation)

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

Registration and administrative information

All participants (onsite and online) must register online.  

Registration deadline is 26th January 2026.  

For further information on registration and payment, please email: Prof. Dr.-Ing. Michael Dumbser (michael.dumbser@unitn.it) Tel. +39 0461 28 2659    

Once the payment has been made, please send a copy of the receipt via email to the address indicated above.

On-site participants are required to bring their own laptops with MATLAB pre-installed.
Online participants will receive the ZOOM link of the course only after payment of the course fee.  

Payment of the course fee must be made only after registration and between January 1st, 2026, and February 2nd, 2026, by bank transfer or credit card.  Full details for payment of the course fee will be sent individually to each participant once the registration process is completed.