Tonatiuh SanchezVizuet
 Assistant Professor, Mathematics
 Member of the Graduate Faculty
 (520) 6216098
 Mathematics, Rm. 115
 Tucson, AZ 85721
 tonatiuh@arizona.edu
Biography
Tonatiuh SánchezVizuet was born in Mexico City. Heearned a “licenciatura”(roughly equivalent to a master’s degree) in Physics and a master’s degree in Mathematics both from the National Autonomous University of Mexico (UNAM). He then earned a Ph.D. in Applied Mathematics from the University of Delaware.
After his doctoral studies, SánchezVizuet spent four years at the Courant Institute of Mathematical Sciences at New York University as a postdoctoral associate. He has been an Assistant Professor of mathematics at the University of Arizona since 2020, where he is also a member of the Graduate Interdisciplinary program in Applied Mathematics. His research interests revolve around the development, analysis, and implementation of numerical methods for the simulation of physical processes. His work has touched the areas of elastic wave propagation, wave scattering, and magnetic equilibrium of plasmas in fusion reactors.
Degrees
 Ph.D. Applied Mathematics
 University of Delaware, Newark, Delaware, United States
 Integral and coupled integralvolume methods for transient problems in wavestructure interaction
 M.S. Mathematics
 National Autonomous University of Mexico (UNAM), Mexico City, Mexico
 Numerical solution of the Euler equations for gas dynamics in 3D
 B.S. Physics
 National Autonomous University of Mexico (UNAM), Mexico City, Mexico
Work Experience
 The University of Arizona, Tucson, Arizona (2020  Ongoing)
 The University of Arizona, Tucson, Arizona (2020  Ongoing)
 Courant Institute of Mathematical Sciences, New York University (2016  2020)
Awards
 National system of researchers (membership level 1 renewal)
 National council for science and technology, Mexico. (CONACyT), Summer 2020
 Lathisms Up and coming Hispanic Mathematician
 Lathisms.org, Fall 2018
 National system of researchers (membership level 1)
 National council for science and technology, Mexico. (CONACyT), Spring 2018
Interests
Research
Numerical Analysis, Scientific Computing, Computational Physics, Applied Mathematical Analysis
Teaching
Partial differential equations, numerical methods, statistics, applied mathematics
Courses
202425 Courses

Honors Thesis
MATH 498H (Fall 2024)
202324 Courses

Directed Research
MATH 492 (Spring 2024) 
Real Analysis
MATH 523B (Spring 2024) 
Directed Research
MATH 492 (Fall 2023) 
Real Analysis
MATH 523A (Fall 2023)
202223 Courses

Anls Ord Diff Equations
MATH 355 (Spring 2023) 
Directed Research
MATH 492 (Spring 2023) 
Intro to Statistical Computing
DATA 375 (Fall 2022) 
Theory of Statistics
MATH 466 (Fall 2022)
202122 Courses

Intro Statistical Method
DATA 363 (Spring 2022) 
Intro Statistical Method
MATH 363 (Spring 2022) 
Independent Study
MATH 599 (Fall 2021) 
Intro Statistical Method
DATA 363 (Fall 2021) 
Intro Statistical Method
MATH 363 (Fall 2021)
202021 Courses

Intro:Stat+Biostatistics
MATH 263 (Spring 2021) 
Calculus I
MATH 125 (Fall 2020)
Scholarly Contributions
Journals/Publications
 Elman, H. C., Liang, J., & SanchezVizuet, T. (2021). Surrogate Approximation of the GradShafranov Free Boundary Problem via Stochastic Collocation on Sparse Grids. Journal of Computational Physics.
 SanchezVizuet, T., Solano, M., & Sanchez, N. (2022). Afternote to Coupling at a distance: convergence analysis and a priori error estimates .. Computational Methods in Applied Mathematics, 22(4), 28. doi:10.1515/cmam20220004
 Solano, M., Sanchezvizuet, T., & Sanchez, N. (2022). Error Analysis of an Unfitted HDG Method for a Class of Nonlinear Elliptic Problems. Journal of Scientific Computing, 90(3). doi:10.1007/s10915022017671
 Sánchez, N., SanchezVizuet, T., & Solano, M. E. (2021). Error analysis of an unfitted HDG method for a class of nonlinear elliptic problems. Journal of Scientific Computing, 90(3), 128. doi:10.1007/s10915022017671
 Elman, H. C., Liang, J., & SanchezVizuet, T. (2021). Surrogate Approximation of the Grad–Shafranov Free Boundary Problem via Stochastic Collocation on Sparse Grids. Journal of Computational Physics, 125. doi:10.1016/j.jcp.2021.110699
 SanchezVizuet, T., & Hsiao, G. C. (2021). Boundary integral formulations for transient linear thermoelasticity with combined type boundary conditions. SIAM Journal of mathematical analysis, 4(53), 3888–3911. doi:10.1137/20M1372834
 SanchezVizuet, T., & Hsiao, G. C. (2021). TimeDependent WaveStructure Interaction Revisited: Thermopiezoelectric Scatterers. Fluids, 6(3). doi:10.3390/fluids6030101
 Sánchez, N., SánchezVizuet, T., & Solano, M. E. (2021). A priori and a posteriori error analysis of an unfitted HDG method for semilinear elliptic problems. Numerische Mathematik. doi:10.1007/s00211021012218More infoWe present a priori and a posteriori error analysis of a high orderhybridizable discontinuous Galerkin (HDG) method applied to a semilinearelliptic problem posed on a piecewise curved, non polygonal domain. Weapproximate $\Omega$ by a polygonal subdomain $\Omega_h$ and propose an HDGdiscretization, which is shown to be optimal under mild assumptions related tothe nonlinear source term and the distance between the boundaries of thepolygonal subdomain $\Omega_h$ and the true domain $\Omega$. Moreover, a localnonlinear postprocessing of the scalar unknown is proposed and shown toprovide an additional order of convergence. A reliable and locally efficient aposteriori error estimator that takes into account the error in theapproximation of the boundary data of $\Omega_h$ is also provided.[Journal_ref: ]
 SanchezVizuet, T., & Hsiao, G. C. (2020). TimeDomain Boundary Integral Methods in Linear Thermoelasticity. SIAM Journal on Mathematical Analysis.
 SanchezVizuet, T., Cerfon, A. J., & Solano, M. E. (2020). Adaptive Hybridizable Discontinuous Galerkin discretization of the Grad–Shafranov equation by extension from polygonal subdomains. Computer Physics Communications.
 Hsiao, G. C., Sanchezvizuet, T., Sayas, F., & Weinacht, R. J. (2019). A timedependent wavethermoelastic solid interaction. Ima Journal of Numerical Analysis, 39(2), 924956. doi:10.1093/imanum/dry016More infoThis paper presents a combined field and boundary integral equation method for solving timedependent scattering problem of a thermoelastic body immersed in a compressible, inviscid and homogeneous fluid. The approach here is a generalization of the coupling procedure employed by the authors for the treatment of the timedependent fluidstructure interaction problem. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The nonlocal boundary problem is analized with Lubich's approach for timedependent boundary integral equations. Using the Laplace transform in terms of timedomain data existence and uniqueness results are established. Galerkin semidiscretization approximations are derived and error estimates are obtained. A full discretization based on the Convolution Quadrature method is also outlined. Some numerical experiments are also included in order to demonstrate the accuracy and efficiency of the procedure.
 SanchezVizuet, T., & Solano, M. E. (2019). A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation. Computer Physics Communications.
 Brown, T. S., Sanchezvizuet, T., & Sayas, F. (2018). Evolution of a semidiscrete system modeling the scattering of acoustic waves by a piezoelectric solid. Mathematical Modelling and Numerical Analysis, 52(2), 423455. doi:10.1051/m2an/2017045More infoWe consider a model problem of the scattering of linear acoustic waves in free homogeneous space by an elastic solid. The stress tensor in the solid combines the effect of a linear dependence of strains with the influence of an existing electric field. The system is closed using Gauss’s law for the associated electric displacement. Wellposedness of the system is studied by its reformulation as a first order in space and time differential system with help of an elliptic lifting operator. We then proceed to studying a semidiscrete formulation, corresponding to an abstract Finite Element discretization in the electric and elastic fields, combined with an abstract Boundary Element approximation of a retarded potential representation of the acoustic field. The results obtained with this approach improve estimates obtained with Laplace domain techniques. While numerical experiments illustrating convergence of a fully discrete version of this problem had already been published, we demonstrate some properties of the full model with some simulations for the two dimensional case.
 SanchezVizuet, T., Greengard, L., Gamba, I. M., Sormani, C., Tao, T., & Payne, K. R. (2018). The Mathematics of Cathleen Synge Morawetz. Notices of the American Mathematical Society, 65(7), 764778. doi:10.1090/noti1706
 SanchezVizuet, T., Sayas, F., Hsiao, G. C., & Weinacht, R. (2018). A TimeDependent WaveThermoelastic Solid Interaction. IMA Journal of Numerical Analysis, 39(2), 924–956.
 Sanchezvizuet, T., & Cerfon, A. J. (2018). Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion. Plasma Physics and Controlled Fusion, 60(2), 025018. doi:10.1088/13616587/aa963aMore infoWe study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudospectral collocation on a grid defined by the zeros of a nonstandard family of orthogonal polynomials called Maxwell polynomials. Taking a onedimensional equation describing energy diffusion due to Fokker–Planck collisions with a Maxwell–Boltzmann background distribution as the test bench for the performance of the scheme, we find that Maxwell based discretizations outperform other commonly used schemes in most situations, often by orders of magnitude. This provides a strong motivation for their use in highdimensional gyrokinetic simulations. However, we also show that Maxwell based schemes are subject to a nonmodal time stepping instability in their most straightforward implementation, so that special care must be given to the discrete representation of the linear operators in order to benefit from the advantages provided by Maxwell polynomials.
 Hsiao, G. C., Sanchezvizuet, T., & Sayas, F. (2017). Boundary and coupled boundary–finite element methods for transient wave–structure interaction. Ima Journal of Numerical Analysis, 37(1), 237265. doi:10.1093/imanum/drw009More infoWe propose timedomain boundary integral and coupled boundary integral and variational formulations for acoustic scattering by linearly elastic obstacles. Well posedness along with stability and error bounds with explicit time dependence are established. Full discretization is achieved coupling boundary and finite elements; Convolution Quadrature is used for time evolution in the pure BIE formulation and combined with time stepping in the coupled BEM/FEM scenario. Second order convergence in time is proven for BDF2CQ and numerical experiments are provided for both BDF2 and Trapezoidal Rule CQ showing second order behavior for the latter as well.
 SanchezVizuet, T., & Cerfon, A. (2017). Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion. Plasma Physics and Cotrolled Fusion.
 SanchezVizuet, T., & Sayas, F. (2017). Symmetric BoundaryFinite Element Discretization of Time Dependent Acoustic Scattering by Elastic Obstacles with Piezoelectric Behavior. Journal of Scientific Computing.
 SanchezVizuet, T., Brown, T., & Sayas, F. (2017). Evolution of a semidiscrete system modeling the scattering of acoustic waves by a piezoelectric solid. ESAIM: Mathematical Modelling and Numerical Analysis.
 SanchezVizuet, T., Sayas, F., Hassell, M., & Tianyu, Q. (2017). A new and improved analysis of the time domain boundary integral operators for the acoustic wave equation. J. Integral Equations Applications.
 SanchezVizuet, T., Sayas, F., & Hsiao, G. (2016). Boundary and coupled boundary–finite element methods for transient wave–structure interaction. IMA Journal of Numerical Analysis.
 SanchezVizuet, T., Sayas, F., & Dominguez, V. (2015). A fully discrete Calderón calculus for the twodimensional elastic wave equation. Computers & Mathematics with Applications.
Proceedings Publications
 Askham, T., Ball, J., Cerfon, A., Freidberg, J., Greenwald, M., ImbertGérard, L., Kim, E., Landreman, M., Lee, J., Majda, A., Malhotra, D., McFadden, G., O'Neil, M., Parra, F., Qi, D., Rachh, M., Ricketson, L., SanchezVizuet, T., Segal, D., & Wilkening, J. (2022).
High Performance Equilibrium Solvers for Integrated Magnetic Fusion Simulations
. In DOE Technical report.
Presentations
 SanchezVizuet, T. (2023, October). A high order solver for the GradShafranov free boundary problem. 56th Annual meeting of the Mexican Mathematical Society. San Luis Potosi, Mexico: Mexican Mathematical Society.