Eleuterio Toro

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


Toro E F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction.Third Edition. Springer-Verlag Berlin Heidelberg, 2009, 721 pages.Riemann Solvers and Numerical Methods for Fluid Dynamics




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Toro E F (Editor). Godunov Methods: Theory and Applications. Edited Review. Kluwer Academic /Plenum Publishers, New York, 2001, 1077 pages.





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Toro E F. Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley and Sons Ltd, Chichester, 2001, 314 pages.





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Toro E F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Second Edition, Springer-Verlag Berlin Heidelberg, 1999, 624 pages.






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Toro E F and Clarke J. F. (Editors). Numerical Methods for Wave Propagation. Kluwer Academic Publishers, Dordrecht, 1998, 385 pages.





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Toro E F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer-Verlag Berlin Heidelberg, 1997, 592 pages.






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Exiliado en Buckingham Palace (en Espanol)

Author: E. F. Toro


Toro E F, Bertolazzi E, Dumbser M and Vignoli G. Springer-Verlag Berlin. Riemann Numerical Methods for Ordinary ad Partial Differential Equations. Heidelberg, Springer, late 2013

Numerical methods for ordinary and partial differential equations are increasingly becoming a must for senior undergraduate courses in sciences, in all engineering courses and obviously in mathematics.

The authors have, for a number of years, taught a course based on the proposed contents, to engineering students of the 4th year in environmental engineering and for the 4th year in telecommunications engineering in the University of Trento, Italy. The contents of the proposed book encompass, in a unified manner, methods for both ordinary and partial differential equations (hyperbolic, parabolic and elliptic). Throughout the book we adopt a modelling and scientific computing approach, emphasizing on modern methods, available computing tools (eg MATLAB, SCILAB, MAPLE) and practical applications. To round each chapter there will be a section with references for further study and, in the case of a numerical chapter, sources of public domain software and their main features will be outlined.

Each chapter closes with a list of carefully selected exercises; the solution to some of these will also be included. The book is essentially self-contained, requiring only a second level of Calculus, basic notions of linear algebra and of scientific programming.

There are two main parts to the book:

Part One deals with numerical methods and Part Two deals with applications presented in the form of case studies. The idea behind is that potential users will come from many, and very diverse, areas in science and engineering. Their applications in mind will therefore be very different but all will have in common the language of mathematics and of numerical methods. The study of the numerical methods will be strongly based on what is common to all potential users. For each class of equations, carefully selected model equations are introduced, which serve as the vehicle for the presentation of the numerical methods. In this manner, the technical complexities associated with the physical meaning, for example, of the equations are eliminated.

Part Two will contain the applications and users could select the particular examples of their specific interest.

E F Toro, Munz C D, Dumbser M and Titarev V. AADER Finite Volume and Discontiunous Galerking Schemes. Springer-Verlag, Berlin Heidelberg, Springer, late 2013

This monograph of about 350 pages is devoted to the study of advanced numerical methods for solving systems of linear and non-linear partial differential equations that model wave propagation phenomena in a variety of media. The basic equations of interest are systems of homogeneous hyperbolic partial differential equations, but extensions containing source terms and high-order spatial derivatives will also be considered. The target readership are researchers on numerical methods for partial differential equations, scientists and engineers concerned with applications that demand the use of advanced numerical methods and lecturers of numerical analysis and scientific computing in universities.

The book will present two main approaches: the finite volume approach and the discontinuous Galerkin finite element approach, both unified via the ADER methodology and the solution of the associated High-Order Riemann Problem. What is distinctive about the numerical methods studied in this book is that they are of arbitrary order of accuracy in both space and time, are applicable to arbitrary grids, typically unstructured grids, and are non-linear, that is, the schemes attempt to circumvent Godunov’s theorem in order to control or avoid spurious oscillations near large gradients.

While for ordinary differential equations, very high order methods have became in practice the usual approach for efficient numerical simulations, for hyperbolic partial differential equations and extensions, this step is still missing. The ADER approach, in the context of two large families of numerical methods (finite volume and discontinuous Galerkin finite elements), is a major step forward in the direction of constructing numerical schemes of arbitrary order of accuracy for hyperbolic partial differential equations and extensions. Recent developments have resulted, for the first time in the field of hyperbolic PDEs, in computer programs in which the (arbitrary) order accuracy is simply a parameter to be prescribed by the user. The proposed monograph assembles material disseminated over a large number of research papers produced, in the main, during the last decade, including a body of contributions from the authors themselves.

The proposed book is seen as a teaching instrument and therefore the presentation of the contents will follow a pedagogical approach. The book reformulates the general framework for constructing numerical methods for hyperbolic PDEs and extensions in a way that the subject becomes simpler and clearer, which would aid its study by the less experienced reader. Given the unavoidable complexity of the advanced numerical methods studied, taking a teaching approach is particularly relevant. We take the view that the studied methods will be implemented, assessed and applied by the user and therefore the emphasis will be on the algorithmic aspects of the methodology, referring the reader to existing research publications for the more complex theoretical justifications.

In addition to the description of the numerical schemes, the book includes case studies in which the methods are applied to solve practical problems in several areas of application: Compressible Fluid Dynamics, Aeroacoustics, Atmospheric Flows, Seismic Waves, Electromagnetics, Tsunami Wave Propagation, Astrophysical problems.

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