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 (1998, 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.


  • 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).


Progress towards a global circulation mathematical model,  incorporating detailed CSF and lymphatics dynamics. Conference of the International Society for Neurovascular Disease held at The New York Academy of Sciences at 7 World Trade Center, New York, April 29th-30th, 2016.


Neurological Diseases and Interacting Fluid Compartments of the Central Nervous System: A Holistic Modeling Approach. Invited plenary lecture given at the University of Pittsburgh USA on occasion of the 5th International Conference on Computational and Mathematical Biomedical Engineering, 10 – 12 April 2017.


Simplified ADER schemes based on a time-reconstruction GRP solver. Conference in honour of Professor Phil Roe’s 80th Birthday, 4th and 5th July 2018. Maxwell Centre, University of Cambridge, UK.


Cranio-spinal fluid dynamics and disease: holistic modelling approach. Conference on Mathematics for BioMedicine. Accademia Nazionale dei Lincei. Rome, Italy, 8-11 October 2018.


Cranio-spinal fluid dynamics and disease: holistic modelling approach. University of Magdeburg, Germany, 22-24 October 2018. Workshop: Modeling and Simulation of the Cardiovascular system.

(Talk given by Dr C. Contarino on my behalf)



Computational medical fluid dynamics: constructing a virtual human. Invited plenary lecture to an audience of about 5000 medical researchers.  ISMRM 27th Annual Meeting & Exhibition, 11-16 May 2019. Palais des congrès de Montréal, 1001 Place Jean-Paul-Riopelle, Montréal, QC, Canada.
SMRT 28th Annual Meeting • 10-13 May 2019

Watch recording: https://youtu.be/qS3EulB7Dko


Part 1. Numerical schemes. Part 2. Brain fluid dynamics and associated diseases: a holistic modelling approach. Invited lecture, CSE Symposium at the University of Toronto, May 21-22, 2019


Meniere’s disease and inner-rear circulation: a computational study. Invited plenary talk at the9th Annual meeting of the International Society of Neurovascular Disease. May 30-31 2019, University of Ferrara., Italy


Simplified ADER schemes based on a time-reconstruction solver for the Generalised Riemann Problem. Contribution to HONOM2019. European Workshop on High Order Numerical Methods for Evolutionary PDEs. Theory and Applications. Madrid, Spain, 1-5 April, 2019.


Godunov methods: the ADER path to high-order of accuracy. Invited lecture at the international conference on Mathematics and its Applications, in honour of the 90th birthday of Professor Sergei K. Godunov. August 4-10, 2019, Novosibirsk, Russia.

On flux splitting schemes for a class of hyperbolic systems. Ninth International Conference on Numerical Methods for Multi-Material Fluid Flow, 9th-13th September 2019. University of Trento, Italy.



The ADER path to high-order schemes for hyperbolic equations. Invited plenary talk at the 18th International Conference on Hyperbolic Problems: Theory, Numerics, Applications. 22-26 June 2020. Malaga, Spain