Numerical Methods

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).

The Remission Project

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

PhD Studies

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.

Research highlights

(Last update: 3rd April 2013)

Here I briefly describe my contributions to research on numerical methods for hyperbolic balance laws that, in my view, have made an impact in the scientific
community and have apparently survived the test of time.

WAF: Weighted Average Flux Method
The WAF numerical flux uses more wave information from the solution of the piecewise constant Riemann problem than the Godunov method (first order), so as to obtain second-order accuracy in space and time, without data reconstruction. Ref [1] presents the idea for one-dimensional systems, while in Ref [2] the WAF method is extended to multidimensional problems on structured meshes.

[1] E F Toro. A Weighted Average Flux method for hyperbolic conservation laws. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. Vol. 423, pages 401-418, 1989.

[2] S J Billett and E F Toro. On WAF-type schemes for multidimensional hyperbolic conservation laws. Journal of Computational Physics. Vol. 130, pages 1-24, 1997.

HLLC: Harten-Lax-van Leer Contact Riemann Solver
The Riemann solver of Harten Lax and van Leer (1983) assumes a two-wave structure, that is a two-wave model. This is the HLL solver. For systems with more than two equations HLL misses the intermediate characteristic fields leading to exceedingly diffusive numerical schemes. The idea of the HLLC scheme is to put back the missing intermediate characteristic fields. For the 3D Euler equations it is enough to restore just one linear field to account for the three fields corresponding to entropy and shear waves. The original idea was first presented in Ref [3], then published in [4] and [5]. There are numerous extensions and improvements to HLLC in the literature.

[3] E F Toro, M Spruce and W Speares. Restoration of the contact surface in the HLL Riemann solver. Technical report CoA 9204. Department of Aerospace Science, College of Aeronautics, Cranfield Institute of Technology. UK, June, 1992.

[4] E F Toro, M Spruce and W Speares. Restoration of the contact surface in the Harten-Lax-van Leer Riemann solver. Shock Waves. Vol. 4, pages 25-34, 1994.

[5] A Chakraborty and E F Toro. Development of an approximate Riemann solver for the steady supersonic Euler equations. The Aeronautical Journal. Vol. 98, pages 325-339, 1994.

FORCE: First Order Centred Flux
This is a centred numerical flux in which no explicit wave propagation information is used, that is the flux uses a 0-wave model. FORCE is derived from a re-interpretation of the staggered-grid version of Glimm’s method, or Random Choice Method. The resulting conservative scheme is non-staggered and has CFL stability limit of unity. The original idea was first presented in Ref [6] and then included in [7]. Convergence is proved in Ref. [8] for two examples of non-linear systems of hyperbolic conservation laws. A variation of FORCE is PRICE: Primitive Centred Schemes. These schemes extend the idea of the FORCE flux to equations written in non-conservative form (no flux) and using the primitive or physical variables. The schemes are suitable for problems with smooth solutions or for systems with weak shocks. See [9]. A major advance is presented in [10], where FORCE is extended to multidimensional
conservation laws solved on unstructured meshes. In Ref [11] FORCE is extended to non-conservative systems of hyperbolic balance laws.

[6] E F Toro. On Glimm-related schemes for conservation laws. Technical Report. Department of Mathematics and Physics. Manchester Metropolitan University, UK. 1996.

[7] E F Toro and S J Billett. Centred TVD schemes for hyperbolic conservation laws. IMA Journal of Numerical Analysis. Vol. 20, pages 47-79, 2000.

[8] G Q Chen and E F Toro. Centred schemes for non-linear hyperbolic equations. Journal of Hyperbolic Equations. Vol. 1, pages 531-566, 2004.

[9] E F Toro and A Siviglia. PRICE: primitive centred schemes for hyperbolic systems. International Journal for Numerical Methods in Fluids. Vol. 42, pages 1263-1291, 2003.

[10] E F Toro, A Hidalgo and M Dumbser. FORCE schemes on unstructured meshes I: conservative hyperbolic systems. Journal of Computational Physics. Vol. 228, pages 3368-3389, 2009.

[11] M Dumbser, A Hidalgo, M Castro, C Pares and E F Toro. FORCE schemes on unstructured meshes II. Non-conservative hyperbolic systems. Computer Methods in Applied Mechanics and Engineering. Vol. 199, Issues 9-12, pages 625-647, 2010.

MUSTA: Multistage Numerical Flux
The MUSTA flux is a centred flux, analogous to FORCE, which attempts to improve the accuracy in the representation of intermediate characteristic fields. This is achieved in a multi-stage fashion using a simple flux at each stage. No wave information is explicitly used. The original idea was first presented in [12] and then published in [13], [14] and [15]. The scheme has been found to be useful for very complex systems of equations, for which sufficiently simple and accurate Riemann solvers do no exist and might never be available.

[12] E F Toro. MUSTA: A multi-stage numerical flux. Isaac Newton Institute for Mathematical Sciences Preprint Series NI04008-NPA. University of Cambridge, UK. Available in pdf format at:

[13] E F Toro. A multi-stage numerical flux. Applied Numerical Mathematics. Vol. 56, pages 1464-1479, 2006.

[14] V A Titarev and E F Toro. MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements. International Journal for Numerical Methods in Fluids. Vol. 49, pages 117-147, 2005.

[15] E F Toro and V A Titarev. MUSTA fluxes for systems of conservation laws. Journal of Computational Physics. Vol. 216, pages 403-429, 2006.

EVILIN: EVolved Initial condition LINearized Riemann solver.
The idea is this: given general initial, piece-wise constant data for the conventional Riemann problem I propose to evolve such data in a single step using a simple conservative method. In this manner large data becomes small data. Then, solve the Riemann for the evolved (small data) initial conditions, for which simple linearizations are justified and lead to close form solutions. The technique is easily applied to complicated hyperbolic systems. See Ref. [16].
[16]. E F Toro. Riemann solvers with evolved initial conditions. International Journal for Numerical Methods in Fluids. Vol. 52, pages 433-453, 2006.

ADER: Arbitrary Accuracy Derivative Riemann Problem Method
ADER represents a major step forward in the design of high-order non-linear numerical methods for hyperbolic balance laws. The schemes are also applicable to parabolic problems. The key idea is to define the local generalized Riemann problem (GRP), solve it and calculate a time integral average to obtain the numerical flux. The form of the scheme is identical to that of the first-order Godunov method, a one-step method. The GRP here is not that in which the initial condition is piece-wise linear (polynomials of degree one). The GRP here is the Cauchy problem in which (i) the initial condition is piece-wise smooth, such as polynomials of any degree for example, and (ii) if source terms are present in the equations, then these are included in the solution of the generalized Riemann problem. I also call this GRP, high-order Riemann problem.

The ADER schemes are fully discrete and use non-linear spatial reconstructions only once per time step. The ADER scheme is identical to the scheme of Harten and collaborators (1987) for the linear advection equation but difers for non-linear problems. The original ADER was presented in [17] for linear systems on Cartesian meshes. The scheme was then extended to non-linear one-dimensional problems in [18] and [19]. Major advances have been made by many collaborators (e.g Munz, Dumbser, Kaeser, Iske, Balsara and Castro) and other researchers. Current developments of the ADER methodology are set in both the finite volume and the discontinuous Galerkin finite element frameworks. See my complete list of publications. In such advances, it is not always easy to recognize the original ADER framework with its two components: non-linear reconstruction (ENO, WENO or other) followed by the solution of the Generalized Riemann problem to compute a numerical flux. The unified ADER framework is presented in [20], in which the various versions of ADER are simply due to the particular way of solving the generalized Riemann problem. For an introduction to ADER schemes see Chapters 19 and 20 of my book,
Ref. [21].

The ADER schemes are beginning to make an impact on several areas of application, such as aero acoustics, seismic wave propagation, tsunami wave propagation and astrophysics.

[17] E F Toro, R C Millington and L A M Nejad. Towards very high-order Godunov schemes. In Godunov Methods: Theory and Applications. Edited Review. E F Toro (Editor). Kluwer Academic/Plenum Publishers. Conference in Honour of S K Godunov. Vol. 1, pages 897-902. New York, Boston and London, 2001.

[18] E F Toro and V A Titarev. Solution of the generalised Riemann problem for advection-reaction equations. Proceedings of the Royal Society of London. Series A. Vol. 458, pages 271-281, 2002.

[19] V A Titarev and E F Toro. ADER: arbitrary high order Godunov approach. Journal of Scientific Computing. Vol. 17, pages 609-618, 2002.

[20] G I Montecinos, C E Castro, M Dumbser and E F Toro. Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms. Journal of Computational Physics. Vol. 231, pp 6472-6494, 2012.

[21] Toro E F. Riemann solvers and numerical methods for fluid dynamics. A practical introduction. 3rd Edition. Springer. Dordrecht, Heidelberg, London and New York, 2009.

DOT: Dumber-Osher-Toro Riemann Solver
A novel method to solve the classical Riemann problem is due to my young colleague Michael Dumbser. Efectively, DOT is a numerical generalization of the OsherSolomon numerical flux. See Refs. [22], [23], [24]. Such generalization has turned out to be very powerful; it is applicable to any hyperbolic system.

[22] Michael Dumbser and Eleuterio Toro. A simple extension of the Osher Riemann solver to general non-conservative hyperbolic systems. Journal of Scientific Computing. Volume 48, pages 70-88, 2011.

[23] Michael Dumbser and Eleuterio Toro. On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Communications in Computational Physics. Vol. 10, pages 635-671, 2011.

[24] E F Toro and M Dumbser. Reformulated Osher-type Riemann solver. Computational Fluid Dynamics 2010. Springer-Verlag, Alexander Kuzmin (editor), 2011, pages 131-136.

NUMERICA: A Library of Source Codes for Teaching, Research and Applications.
Free Software.
NUMERICA is a library of 50 source codes for solving hyperbolic partial differential equations using a broad range of modern, high resolution shock-capturing numerical methods. There are three sub-libraries: HYPER-LIN, HYPER-EUL and HYPER-WAT, see detailed references below, Refs. [25], [26] , [27] .The library is publicly available from my website.

[25] E F Toro. NUMERICA: A Library of Source Codes for Teaching, Research and Applications. HYPER-LIN. Methods for model hyperbolic equations. Numeritek Limited UK. ISBN 0-9536483-0-3. Software. 1999.

[26] E F Toro. NUMERICA: A Library of Source Codes for Teaching, Research and Applications. HYPER-EUL. Methods for the Euler equations. Numeritek Limited UK. ISBN 0-9536483-3-8. Software. 1999.

[27] E F Toro. NUMERICA: A Library of Source Codes for Teaching, Research and Applications. HYPER-WAT. Methods for the shallow water equations. Numeritek Limited UK. ISBN: 9536483-5-4. Software. 2000.