Center for Computing Research (CCR)

Center for Computing Research

Ignacio Tomas

Ignacio Tomas
Computational Mathematics
Email: itomas@sandia.gov
Phone: 505/844-2667

Mailing address:
Sandia National Laboratories
P.O. Box 5800, MS 1320
Albuquerque, NM
87185-1320

My main interests are PDEs and their respective discretization using methods that preserve essential “structural stability properties”, in particular: energy estimates, entropy inequalities, invariant sets, maximum/minimum principles, asymptotic limits, and pointwise positivity properties of the original PDE. One of my core interest lies in PDEs that preserve two or more of such notions of stability, in particular, when energy-stability is either non-applicable or merely a complementary notion of stability. The vast majority of PDE models having actual physical/technical application fall in such category.

Influenced by my work within a research group with strong focus on multifluid plasma models, I am particularly interested in Euler-like systems such as Euler-Maxwell and Euler-Poisson. Euler systems (in the broadest sense possible) are pervasive in mathematical physics: from self-gravitation, relativistic particle beams, to quantum mechanics. For instance the Schrodinger-Poisson system can be rewritten as the Quantum-Euler-Poisson system using Madelung's transform, leading to an entire set of applications that goes well beyond traditional continuum physics and shock-hydrodynamics. Systems such as Euler-Maxwell and Euler-Poisson do not have an evident (or explicit) variational structure that we can exploit numerically: the operator in space satisfies neither positivity, nor coercivity, nor Garding inequality, nor inf-sup inequality, nor related dissipative property. Boundedness in-norm cannot be expected to hold true in such context and L2-minimizers of the PDE-residual are unlikely to characterize the right notion of solution. This lack of an explicit variational structure is, to large extent, an early indicator that Galerkin techniques are unlikely to yield an outcome of long lasting value. This forces me to think about numerical schemes in terms of PDE properties rather than fabricated notions of stability.

The departure from the traditional energy-like framework (Galerkin methods) is by no means a novel idea. Such change of priorities has already taken place in the PDE literature many decades ago with the introduction of entropy-solutions and viscosity-solutions in various settings. In this regard, Ansgar Jungel's book "Entropy Methods for Diffusive Partial Differential Equations" is particularly interesting, showing that in general we have consider a lot more than just energy-stability, even for PDEs of dissipative/parabolic nature. The challenge lies in the development of finite dimensional counterparts of such properties that can effectively be materialized as computer code.

Education/Background

Visiting Assistant Professor, 2015-2018
Department of Mathematics,
Texas A&M University, College Station.
Advisor/s: Jean-Luc Guermond, Bojan Popov.

Ph.D. in Applied Mathematics, 2015.
University of Maryland College Park (UMCP).
Thesis: Ferrofluids: modeling, numerical analysis, and scientific computation.
Advisor: Prof. Ricardo H. Nochetto.

M.Sc. in Applied Mathematics, 2012.
University of Maryland College Park (UMCP).
Advisor: Prof. Ricardo H. Nochetto.

Mechanical Engineer, 2005.
Universidad Nacional de Mar del Plata (UNMdP), Argentina.

Selected Publications & Presentations

2020
  • Tomas, Ignacio, Daniel Arndt, Wolfgang Bangerth, Bruno Blais, Timo Heister, Thomas C Clevenger, Luca Heltai, Matthias Maier, Jean-Paul Pelteret, Reza Restak, Bruno Turcksin, "The deal.II Library, Version 9.2," Conference Paper, Journal of Numerical Mathematics, June 2020.
2019
  • Tomas, Ignacio, Jean-Luc Guermond, Bojan Popov, "Invariant domain preserving schemes for hyperbolic systems of conservation laws," Presentation, Multimat 2019 in Trento, , Italy, September 2019.
2018
  • Tomas, Ignacio, Jean-Luc Guermond, Bojan Popov, Murtazo Nazarov, "Second-Order Invariant Domain Preserving Approximation of the Euler Equations Using Convex Limiting," Journal Article, SIAM Journal on Scientific Computing, Vol. 40, No. 5, pp. A3211–A3239, Accepted/Published October 2018.