Dr. Gerardo Hernandez-Duenas
  Associate Professor
  Institute of Mathematics
  Unit Juriquilla
  UNAM
 
  Office: 2
  Institute of Mathematics
  UNAM Campus Juriquilla
  Blvd Juriquilla # 3001
  76230 Juriquilla, Querétaro, México
  Phone: (52) 442-192-6287
  Email: hernandez at im dot unam dot mx
 
 
 
 
 
I did my undergraduate studies at the Departamento de Matemáticas - Universidad de Guanajuato under the supervision of my undergraduate thesis adivser Xavier Gómez-Mont Ávalos. I obtained my Ph. D. degree at the Department of Mathematics at The University of Michigan - Ann Arbor. My Ph. D. advisers were Alejandro Uribe and Smadar Karni. I was a Van Vleck visiting scholar (Posdoc position) at the Department of Mathematics at the University of Wisconsin - Madison. I worked with my posdoc advisers Leslie M. Smith and Samuel N. Stechmann.
 
 
 
 

Research areas:

  • Applied Mathematics and Atmospheric Sciences

  • Numerical Analysis and Hyperbolic Conservation Laws

  • Geophysical Fluid Dynamics

  • Semiclassical Analysis

  • Turbulence

 
 
I am interested in both theory and numerics of mathematical models based on Partial Differential Equations. In the theoretical part, I study models derived from physical laws with applications to Atmospheric Sciences, Geophysical Fluid Dynamics and Physics. A variety of models can be derived from the Navier-Stokes equations or similar by assymotic arguments or by assuming simplifying conditions. Such models are to maintain a balance between reality and model complexity. Examples include: shallow water equations, blood flow models, Boussinesq equations, quasi-geostrophic equations, anelastic equations and simplifications to model precipitating turbulent convection, among others.
 
In Mathematical Physics, I am also interested in Schrodinger-type operators. In particular, I work on techniques developed in Semiclassical Analysis for Pseudodifferential operators.
 
In numerical analysis, I am interested in developping efficient numerical methods to implement a variety of models. Models in the category of Hyperbolic Conservation laws may encounter numerical difficulties achieving stability or reaching convergence to the entropy-satisfying solutions in the presence of shock waves. In Geophysical Fluid Dynamics, one may need to use different techniques such as pseudo-spectral methods to analyze different types of waves interacting in an atmospheric phenomenon and ensuring that the solution satisfies the continuity equation when appropriate, etc.
 
Erdős number: 4
Me --- Andrew J. Majda --- Charles Louis Fefferman --- J. Marshall Ash --- Paul Erdős1
 
 

 

 

Complete list of papers (last updated 2020/02/07):

 

12. G. Hernandez-Duenas, S. Stechmann and L. Smith. Weak- and Strong- Friction Limits of Parcel Models: Comparisons and Stochastic Con- vective Initiation Time. Quarterly Journal of the Royal Meteorological Society (2019).

11. G. Hernandez-Duenas, Ulises Velasco-García and Jorge Velasco Hernández. A positivity-preserving central-upwind scheme for isentropic two- phaseflows through deviated pipes. ESAIM: Mathematical Modelling and Numerical Analysis(2019).

10. G. Hernandez-Duenas. A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients. Journal of Scientific Com- puting(2017), Volume 73 (2-3) pp 753-782. 

9. G. Hernandez-Duenas and Abdelaziz Beljadid. A central-upwind scheme with artificial viscosity for shallow-water flows in channels. Advances in Water Resources 96 (2016) 323-338.

Paper

Link to Journal

Figure: Schematic of channel in numerical experiment of dam break

 

 

8. G. Hernandez-Duenas, Leslie M. Smith, and Samuel N. Stechmann. Stability and instability criteria for idealized precipitating hydrodynamics. Journal of Atmospheric Sciences, Vol 72, No. 6 (2015), pp. 2379-2393.

Paper

Link to Journal

How to cite this article

Figure: Growth rates versus horizontal wavenumbers for different values of rainfall speed.

 

7. G. Hernandez-Duenas and Alejandro Uribe. Algebras of semiclassical pseudodifferential operators associated with Zoll-type domains in cotangent bundle. Journal of Functional Analysis, 268 no. 7 (2015), pp. 1755-1807.

Paper

Link to Journal

How to cite this article

Figure: Propagation of a coherent state in a Zoll-type domain.

 

6. Jorge Balbás and G. Hernandez-Duenas.  A Positivity Preserving Central Scheme for Shallow Water Flows in Channels with Wet-Dry States. ESIAM: Mathematical Modelling and Numerical Analysis (M2AN) 48 (2014) 665-696.

Paper

Link to Journal

Figure: Dam break simulati

Smadar Karni and G. Hernandez-Duenas.

Hyperbolic Problems: Theory, Numerics, Applications.


Proceedings of Simposia in Applied Mathematics, Volume 67, Part 2, (2009), 715-724.


Amer. Math. Soc., Providence, RI, (2009).

Paper

Link to Journal

How to cite this article
Figure: Computed and exact solutions in a shock-tube problem using a conservative (left) and a hybrid (right) formulation.

 
  Complete list of publications (last updated 2023/July/12):

 

A list of citations can be found here.

Edited books:

 
1. G- Hernandez-Duenas & M. A. Moreles. Mathematical and Computational Models of Flows and Waves in Geophysics. Springer (Book) (1), pp 1-195 (2022).
https://doi.org/10.1007/978-3-031-12007-7
 
 
 
 
 
 
Figure: Book about Geophysical applications.  
 
 

Published papers:

22. Weakly compressible two-layer shallow water flows along channels.
S. Shah & G. Hernández-Dueñas.
Journal of Scientific Computing (2024)
Link to Journal

 

 

 

 

Figure: Two-layer flow in channels.

 
21. Exploring the excess of cloud condensation nuclei and rain suppression using a minimal three-dimensional Boussinesq model with bulk cloud microphysics.
O. Guerrero-Medina & G. Hernández-Dueñas.
Physics of Fluids (2024)
Link to Journal

 

 

 

 

Figure: Graph of aerosols & rain.

20. Global well-posedness of a model for precipitating convection with hydrostatic pressure under fast autoconversion and rain evaporation conditions.
Temporary free access: https://authors.elsevier.com/c/1iUbN,WNxuZyK
Néstor A. Sánchez-Goycochea & G.- Hernández-Dueñas.Chi
Journal of Mathematical Analysis and Applications (2024)
Link to Journal

 

 

 

Figure: Graph of transition function from saturated to unsaturated regions.

 
19. A patch in time saves nine: Methods for the identification of localised dynamical behaviour and lifespans of coherent structures.
C. Blachut, C. González-Tokman & G.- Hernández-Dueñas.
Journal of Nonlinear Science (2023)
Link to Journal
 
 
 
 
 
 
 
 
Figure: Identification of coherent structures using dynamical systems
 
18. A new two-dimensional blood flow model with arbitrary cross sections.
C.A. Rosales-Alcantar & G.- Hernández-Dueñas
ESAIM: Mathematical Modeling and Numerical Analysis (2023).
Link to Journal
 
 
 
 
 
 
 
 
 
Figure: Circulation pattern in blood flow simulation
 
 
17. Bathymetry and friction estimation from transient velocity data for one-dimensional shallow water flows in open channels with varying width.  
G. Hernandez-Duenas, M.A. Moreles  & P. Gonzalez-Casanova
Physics of Fluids (2023)
Link to Journal
 
 
 
Figure: Bathymetry estimation with the steepest descent method
 
16. Perturbations of the Landau Hamiltonian: Asymptotics of Eigenvalue Clusters. 
G. Hernandez-Duenas, S. Pérez-Esteva, A. Uribe & C. Villegas-Blass
Ann. Henri Poincaré (2022). https://doi.org/10.1007/s00023-021-01092-7
 
 
 
Link to Journal 
Figure: Diagram of different limiting regimes for the Landau problem
 
 
15. Impact of Wave-Vortical Interactions on Oceanic Submesoscale Lateral Dispersion.
G. Hernandez-Duenas, M. Pascale Lelong, and Leslie M. Smith
Journal of Physical Oceanography (2021).
 
The numerical model simulations upon which this study is based are too large to archive. The velocity and density data at 200 inertial periods as well as the data for the particles' trajectories from 200 to 210 inertial periods are available here.  
 
Link
Figure: Horizontal contours of linear potential vorticity.
 
14. A central-upwind scheme for two-layer shallow-water flows along channels with arbitrary geometry.
G. Hernandez-Duenas & J. Balbas
ESAIM: Mathematical Modeling and Numerical Analysis (2021).
 Link
 
 
 Figure: Schematic of a two layer shallow water flow
 
13. A well-balanced positivity-preserving central-upwind scheme for one-dimensional blood-flow models.
G. Hernandez-Duenas, G. Ramirez-Santiago
International Journal for Numerical Methods in Fluids (2020). https://doi.org/10.1002/fld.4887
 
Link to Journal
 
Figure: Schematic for blood-flow model 
 
 12. Weak- and Strong- Friction Limits of Parcel Models: Comparisons and Stochastic Convective Initiation Time.  G. Hernandez-Duenas, S. Stechmann and L. Smith. Quarterly Journal of the Royal Meteorological Society (2019), Volume 145, No. 722, pp 2272-2291  (2019). 
Link to Journal 

 

Figure: Bistability perspective of conditional stability

 
11. A positivity-preserving central-upwind scheme for isentropic two- phaseflows through deviated pipes.
G. Hernandez-Duenas, Ulises Velasco-García and Jorge Velasco Hernández. ESAIM: Mathematical Modelling and Numerical Analysis (2019). 
Link to Journal 

Figure: Schematic of the two-phase flows in deviated pipes.

 10. A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients.  G. Hernandez-Duenas. Journal of Scientific Computing (2017), Volume 73 (2-3) pp 753-782.
 Link to Journal   
 
 
 
Figure: Radial dam break over flat topography (height, density and pressure)
 
9. A central-upwind scheme with artificial viscosity for shallow-water flows in channels.
G. Hernandez-Duenas and Abdelaziz Beljadid.
Advances in Water Resources 96 (2016) 323-338.
Link to Journal
 
 
 
Figure: Schematic of channel in numerical experiment of dam break.
 
8. Stability and Instability Criteria for Idelized Precipitating Hydrodynamics.
G. Hernandez-Duenas, Leslie M. Smith, and Samuel N. Stechmann.
Journal of Atmospheric Sciences, Vol 72, No. 6 (2015), pp. 2379-2393
Link to Journal
 
 
Figure: Growth rates versus horizontal wavenumbers for different values of rainfall speed. 
 
7. Algebras of semiclassical pseudodifferential operators associated with Zoll-type domains in cotangent bundles
G. Hernández-Dueñas and Alejandro Uribe.
Journal of Functional Analysis, 268 no. 7 (2015), pp. 1755-1807
Link to Journal
 
 
 
Figure: Propagation of a coherent state in a Zoll-type domain.
 
6. A Positivity Preserving Central Scheme for Shallow Water Flows in Channels with Wet-Dry States.
Jorge Balbás and G. Hernández-Dueñas. 
ESIAM: Mathematical Modelling and Numerical Analysis (M2AN) 48 (2014) 665-696.
Link to Journal
 
Figure: Dam break simulation at different times.
Blue: Water height. Brown: Bottom topography. Gray: Walls.
 
 
5. Investigation of Boussinesq dynamics using intermediate models based on wave-vortical interactions.
G. Hernandez-Duenas, Leslie M. Smith, and Samuel N. Stechmann
Journal of Fluid Mechanics, 747 (2014), 247-287
Link to Journal
 
 
 
 Figure: Dipole evolution given by the Boussinesq model
 
4.  Minimal models for precipitating turbulent convection
G. Hernandez-Duenas, Andrew J. Majda, Leslie M. Smith, and Samuel N. Stechmann.
Journal of Fluid Mechanics, 717 (2013), 576-611.
 
Link to Journal
 
 
 
Figure: Contours of rain water. Scattered convection (left) versus squall lines (right).
 
3. Shallow Water Flows in Channels.
G. Hernández and Smadar Karni.
J. Sci. Comput. 48 (2011), no. 1-3, 190-208.
Link to Journal
 
 
Figure: Exact and numerical steady state (discontinuous transcritical) solutions to shallow water.
 
2. A Hybrid Algorithm for the Baer-Nunziato Model Using the Riemann Invariants.
Smadar Karni and G. Hernández-Dueñas.
J Sci Comput, 45, (2010), 382-403.
 
Link to Journal
Figure 1: Schematic: Gas flow over a porous particle bed.

1.A Hybrid Scheme for Flows in Porous Media.
Smadar Karni and G. Hernández-Dueñas.
Hyperbolic Problems: Theory, Numerics, Applications.
Proceedings of Simposia in Applied Mathematics, Volume 67, Part 2, (2009), 715-724.
Amer. Math. Soc., Providence, RI, (2009).

Link to Journal

Submitted papers:

 1.  S. Shah & G.- Hernandez-Duenas. A weakly-compressible two-layer shallow water model in general channels.
 
2. O. Guerrero-Medina & G. Hernandez-Dueñas. Exploring the excess of cloud condensation nuclei and rain suppression using a minimal 3D Boussinesq model with bulk cloud microphysics.

Papers in preparation:

1. Jeffrey J. Early, Gerardo Hernández-Dueñas, Leslie M. Smith & M.-Pascale Lelong. Exact expressions for available potential energy and available potential vorticity.

Extended abstracts:

A Scheme for Shallow Water Flow with Area Variation.
Smadar Karni and G. Hernández-Dueñas.
American Institute of Physics.
AIP Conference Proceedings
International Conference on Numerical Analysis and Applied Mathematics
Rethymno, Crete, Greece, 18-22 September 2009. 1168 (2009), 1433-1436.

Link to Journal