Short Course – Computational Methods for Hyperbolic Equations with Applications

Sunday 23rd to Thursday 27th September, 2018

BellaVista Relax Hotel, Levico Terme, Trento, Italy

Prof. Dr. Eleuterio TORO




Brief description

This course is primarily aimed at Master and PhD students in mathematics, engineering, computer science, physics, environmental disciplines, astrophysics and biomedicine, amongst others.

The contents will also be of interest to post-doctoral fellows in academia and industry, as well as to academics involved in teaching numerical methods for partial differential equations or supervising numerically oriented projects.

This course is introductory, with strong emphasis on clear understanding of underlying concepts and principles involved in the numerical approximation of hyperbolic partial differential equations.  Applications will focus on the shallow water equations, the blood flow equations and the Euler equations of compressible gas dynamics.

The course will combine lectures on the basic theory of hyperbolic equations and numerical methods, with practical activities involving exercises by hand and computer-based experimentation using MATLAB programs provided with the course material. Participants are expected to bring their own laptops equipped with MATLAB.


These will include:

  • Theory of linear and non-linear hyperbolic equations
  • Basic numerical analysis: monotonicity,  linear stability and local truncation error
  • Godunov’s theorem and the need for non-linear methods
  • Finite volume methods for hyperbolic balance laws
  • Riemann solvers and centred numerical fluxes for finite volume and DG methods
  • Treatment of source terms and viscous terms
  • Non-linear high order methods based on TVD and ENO reconstructions
  • The shallow water equations and one and two space dimensions
  • The blood flow equations for arteries and veins
  • The compressible Euler equations in one and multiple space dimensions
  • Overview of the software library NUMERICA
  • Introduction to high-order ADER methods

At the end of the course, participants will be equipped with the necessary tools to implement the methods for the applications of their interest and to make, independent fruitful use of modern textbooks and research papers on the subject.

The bases of this course have a long history, going back to the 80s in Cranfield, the 90s in Manchester and then Cambridge, Oxford Mississippi, Tohoku,  Barcelona,  Santiago de Compostela, Cordoba,  Madrid, Moscow, Prague,  Torino, Bologna, Zurich and Trento, amongst others.


Students: € 1300; post-doctoral fellows:  €1500; academics: €1800; others: €2000. The fees will include morning coffee, lunch, afternoon coffee, the course gala dinner on Wednesday 26th September, printed lecture notes and demonstration programs.  Participants are expected to make their own accommodation arrangements, for which helpful  information can be provided on demand.


Details  on the registration procedure and joining instruction will be available soon from:


Nearest airports are Venice and Verona. Train services to Trento and from here to Levico Terme are available. From Venice airport there is also a direct train to Levico Terme through a very picturesque route.

Further information

For further information please write directly to Prof. Toro, email:

For updates please visit website:

The venue for the course

BellaVista Relax Hotel, Levico Terme, Trento, Italy. This is a four start hotel in the picturesque  alpine town of Levico Terme, right next to the Levico Lake.  Participants are encouraged to make their own early accommodation booking directly with the hotel, with whom special arrangements have been made:

BellaVista Relax Hotel
Viale Vittorio Emanuele III, 7
38056 Levico Terme (Tn), Italy
+39 0461 706136
+39 0461 706474