Eleuterio Toro

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



My main interest is in the design of new numerical algorithms of the finite volume and discontinuous Galerkin type for solving partial differential equations in general and hyperbolic balance laws in particular.  Physical issues of relevance in the design include time dependency; strong wave interaction; discontinuous waves such as shocks, contact discontinuities, vortices, material interfaces; long-time evolution and sharp fronts.  Numerical issues of relevance include non-linear schemes (to circumvent Godunov’s theorem) of very high order of accuracy in both space and time.

  • Riemann Solvers

Several numerical methodologies make use of the solution of the Riemann problem; these include Godunov-type methods, the Random Choice Method of Glimm, Front-Tracking Methods, the Discontinuous Galerkin Finite Element Method and, more recently, the Smooth Particle Hydrodynamics (SPH) approach.

I am interested in the construction of approximate Riemann solvers for (a) the equations governing the dynamics of compressible materials with general equations of state, such as gases and water; (b) incompressible viscous flows and (c) for free-surface shallow fluids. Approaches include (i) linearisations based on non-conservative variables (ii) two-shock type approximations (iii) the HLLC approach, a variant of the HLL approach in which linear fields such as contacts are included in the structure of the solution of the Riemann problem, and (iv) adaptive Riemann solvers.

 Recent developments include (a) a generalization of the Osher solver, called the DOT Riemann solver (for Dumbser-Osher-Toro), 2011; (b) new flux vector splitting methods for the Euler equations, 2012; (c) extensions of the centred flux called FORCE, which are applicable to multidimensional problems on unstructured meshes, 2010, 2011.

  • Finite Volume and discontinuous Galerkin methods

My main interest here lies in the design on finite volume schemes of the upwind type and centred type, in which approximate Riemann solvers are used to provide intercell numerical fluxes.

I am also interested in centred schemes, in which numerical fluxes do no make explicit use of wave propagation information.

  • Primitive variable schemes

Conservative methods are mandatory for flows containing shock waves but produce anomalous solutions for special but important cases. Non-conservative schemes avoid some of these difficulties.

I am interested in the design of high-resolution non-oscillatory upwind schemes that are based on primitive variables, as well as schemes that make adaptive use of conservative and non-conservative schemes.

  • Very-high order schemes of the ADER type

There are several areas that are potential beneficiaries of numerical methods that are (i) of very high order of accuracy (eg 10) in both space and time (ii) and are free from spurious oscillations in regions where the solution exhibits large gradients, such as in the vicinity of shock waves and sharp fronts for instance. Godunov’s theorem (1959) says that this is an impossible task if the schemes are linear. Progress has been made in circumventing Godunov’s theorem by constructing non-linear schemes, even when applied to linear problems.

I have taken a new approach for constructing schemes satisfying requirements (i) and (ii) above. We call the approach: ADER. The ideas go back to 1992 and are based on a modified version of the GRP (Generalised Riemann problem) scheme of Ben-Artzi and Falcovitz (1984). The first successful results were reported (with Richard Millington and Lida Nejad, UK) in 2001, where the ADER schemes were completely formulated for linear hyperbolic equations with constant coefficients in 1, 2 and 3 space dimensions; practical implementations of the ideas included results for 1D and 2D problems for schemes of order 10 in space and time.

The joint works with Vladimir Titarev (Russia)  resulted in the extension of the ADER schemes to non-linear systems in multiple space dimensions.  Collaboration with Michael Dumbser resulted in the extension of the ADER schemes to multiple space dimensions on unstructured meshes in the frameworks of finite volume and discontinuous Galerkin approaches.  Several other extensions  and applications of the ADER approach have been reported in the last ten years of so.

Such works are the result of collaboration with many scientists in Europe, USA, China and Japan. These include hyperbolic balance laws with stiff source terms; parabolic equations such as diffusion-reaction equations, the compressible Navier-Stokes equations; compressible two-phase flows and many more. See my complete list of publications for more information.

  • Cartesian-Cut Cell Methods

Generating meshes for complicated computational domains is a difficult task. Cartesian-cut cell methods offer an approach that is gaining increasing popularity. The technique has been applied to simulate the flow around complete helicopter configurations including many ‘extra’ components, in USA and Japan.

My own research concerns approaches to retain as much as possible of the original geometries without ‘cutting corners’ and allowing for cells that are split by solid boundaries. There are many interesting aspects of computational geometry that are common to several other areas in applied and computational mathematics.

There are also issues concerning numerical schemes for cells in the vicinity of domain boundaries such as stability, conservation, treatment of viscous terms and space and time accuracy.

  • Some Applications

I am also interested in the practical implementation of numerical schemes and in their application to problems related to science, engineering and other disciplines such as environmental fluid dynamics. Application areas include:

– Shock waves in gases and liquids

– Combustion-driven waves in gases and high energy solids

– dam-break problems,

– tsunami wave propagation,

– bore reflection patterns in two-dimensional shallow water flows (Mach reflection)

– pollution transport,

– debris flow

– heavy-gas dispersion.

– Blood flow

– Medical problems

– Propulsion technology

  • Multiphase Flows

There are many areas in science and modern technology in which multiphase flow models are used. My own experience is related to compressible, reactive multiphase flows in propulsion technology, in which complex moving boundaries are present. Past and current work in this area has been funded by the British Ministry of Defense via DERA.

Another area of application of my interest is in Nuclear Reactor Safety and Design. This was a topic of collaboration funded by the European Union and involved nine European partners. Work in this area involves myself, Professor E Romenski (Novosibirsk, Russia), Dr A Slaouti and PhD student Dia Zeidan.

There are many research issues here. These include the mathematical character of the equations (elliptic/hyperbolic for most models in use; conservative or non-conservative form) and the development of accurate numerical methods of the type successfully developed for single-phase flows. More recent works in this area involve collaboration with Michael Dumbser (Trento), Svetlana Tokareva (Zurich), Cristobal Castro (Hamburgh); Bok Jik Lee (Cambridge) and Baolin Tian (China).


REMISSION: a long-term research project on Research into Mathematical modelling of Multiple Sclerosis and its vascular connection

Professor Eleuterio Toro

(First version: 7th October 2011; last update: 2nd September 2012)

This long-term research project, funded from various sources,  is concerned with a theoretical study of the potential connection between the venous anomaly called CCSVI (for Chronic Cerebrospinal Venous Insufficiency, discovered by the Italian medical researcher Paolo Zamboni)  and Multiple Sclerosis.

Research team:

Prof. E F Toro (principal investigator), Lucas Mueller (PhD student), Gino Montecinos (PhD student), Laura Facchini (PhD student), Alfonso Caiazzo (Post-Doctoral fellow).

Related collaborators:

Prof. Alberto Bellin (Trento); Prof. Renzo Antolini (Trento); Prof. Michael Dumbser (Trento); Prof. Vincenzo Casulli (Trento); Dr. Annunziato Siviglia (Trento); Prof. Mark Haacke (Detroit), Prof. Paolo Zamboni (Ferrara).

Vascular Theory of Multiple Sclerosis and Mathematics

Recent advances in the vascular theory of Multiple Sclerosis opens up a huge and very challenging field of research for applied and computational mathematicians. In the long-term this very timely research programme aims at constructing a Human Circulation Simulator to theoretically study and discern the controversial hypothesis that anomalous venous blood from the central nervous system leads ultimately to the onset of this disabling so far incurable disease.


Computer simulation of blood flow in the intra/extra cranial venous system in humans with multiple sclerosis and the CCSVI condition.

Summary and aims of the research programme

This research programme is motivated by the recently proposed association between multiple sclerosis (MS) and a vascular anomaly termed chronic cerebro-spinal venous insufficiency (CCSVI) by Zamboni and collaborators. The CCSVI condition is characterized by the presence of obstructions of various kinds in the extracranial veins. Such obstructions prevent a normal drainage of blood from the brain to the heart. CCSVI is present in a relevant number of MS patients and such occurrence is of great clinical interest.

However such association does not yet explain the gestation of MS, although it has been hypothesized a potential link between the altered fluid dynamics, transport and deposition of iron, disruption of the brain- blood barrier and penetration of auto-aggressive immune cells into the CNS, with the known consequences of demyelization of the nerve’s sheath.

CLICK HERE for more information



Call for candidates:

We seek candidates to carry out original research leading to the PhD degree. Funding is available on competitive grounds for candidates of any nationality.


– applied mathematics

– scientific computing

– numerical analysis of partial differential equations

– engineering with a sufficient background in mathematics and mechanics

– physics

– computer science

– medicine

 For further information from Professor E F Toro, send an EMAIL now.