Eleuterio Toro

BSc, MSc, PhD, DHC, OBE. Professor of Numerical Analysis

Personal Profile


My research has for many years focused on  the construction of computational methods for solving partial differential equations, with particular emphasis on hyperbolic balance laws. Contribution highlights include: WAF (1989), HLLC (1992), FORCE (1996),  ADER (2001), MUSTA (2004), TV flux splitting (2012). The ADER framework allows the construction of one-step, non-linear numerical methods  (in the sense of Godunov’s Theorem) of arbitrary order of accuracy in both space and time.  Very high order methods have a huge efficiency gain; such feature increases the power of mathematical modelling and simulation for solving practical problems in science and engineering. The ADER numerical techniques have been found to be relevant in many application areas; examples include  astrophysics, aero-acoustics, shock wave physics, aerospace sciences, seismology, tsunami wave propagation, meteorology, biology  and medicine. More recent contributions include relaxation procedures (2014), whereby general advection-diffusion-reaction equations may be formulated as hyperbolic balance laws with stiff source terms. 

I am currently focused on the study of the physics of neurodegenerative diseases and their association to the disturbed dynamical interaction of materials in the central nervous system (arterial blood, venous blood, interstitial fluid, cerebrospinal fluid, the brain parenchyma and the newly discovered brain lymphatic system). Of  particular interest is the theoretical elucidation of the hypothesised link between extra-cranial venous anomalies and pathologies, such as Multiple Sclerosis, Meniere’s disease and Parkinson’s disease. We are making progress on these very timely themes of practical relevance to so many people around the world. Advances are possible thanks to the contribution of research students, post-doctoral fellows, colleagues in Trento and collaboration with academics and medical doctors in Europe, USA and elsewhere.

Some of our latest results on this topic were presented as an invited plenary talk entitled “Progress towards a global circulation mathematical model,  incorporating detailed CSF and lymphatics dynamics”, at the next conference of the International Society for Neurovascular Disease to be held at The New York Academy of Sciences at 7 World Trade Center, New York, April 29th-30th, 2016.

Next April, at the University of Pittsburgh USA, I will give an invited plenary lecture entitled “Neurological Diseases and Interacting Fluid Compartments of the Central Nervous System: A Holistic Modeling Approach” on occasion of the 5th International Conference on Computational and Mathematical Biomedical Engineering, 10 – 12 April 2017. For details see


My teaching activities include the following  courses:

  • Computational Haemodynamics for the master in mathematics with specialism in “Modelling and simulation for biomedical problems”
  • Advanced numerical methods for hyperbolic equations and applications, offered to PhD students in engineering, mathematics and other scientific disciplines. This intensive short course is open to participants from anywhere in the world.


  • Teacher of Primary Education (Escuela Normal de Victoria, Chile, 1967).
  • Pedagogy in Mathematics (Universidad Catolica de Temuco, Chile).  Studies interrupted in the 5th and final year, in September 1973.
  • BSc Honours in Pure Mathematics (University of Warwick, Great Britain, 1977).
  • MSc in Applicable Mathematics (University of Dundee, Great Britain, 1978).
  • PhD in Computational Mathematics (University of Teesside, Great Britain, 1982).