Tonatiuh Sanchez-Vizuet

Interests

Research

Numerical Analysis, Scientific Computing, Computational Physics, Applied Mathematical Analysis

Teaching

Partial differential equations, numerical methods, statistics, applied mathematics

Courses

Scholarly Contributions

Journals/Publications

• Elman, H. C., Liang, J., & Sanchez-Vizuet, T. (2021). Surrogate Approximation of the Grad-Shafranov Free Boundary Problem via Stochastic Collocation on Sparse Grids. Journal of Computational Physics.
• Sanchez-Vizuet, 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/cmam-2022-0004
• Solano, M., Sanchez-vizuet, T., & Sanchez, N. (2022). Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems. Journal of Scientific Computing, 90(3). doi:10.1007/s10915-022-01767-1
• Sánchez, N., Sanchez-Vizuet, T., & Solano, M. E. (2021). Error analysis of an unfitted HDG method for a class of non-linear elliptic problems. Journal of Scientific Computing, 90(3), 1-28. doi:10.1007/s10915-022-01767-1
• Elman, H. C., Liang, J., & Sanchez-Vizuet, T. (2021). Surrogate Approximation of the Grad–Shafranov Free Boundary Problem via Stochastic Collocation on Sparse Grids. Journal of Computational Physics, 1--25. doi:10.1016/j.jcp.2021.110699
• Sanchez-Vizuet, 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
• Sanchez-Vizuet, T., & Hsiao, G. C. (2021). Time-Dependent Wave-Structure Interaction Revisited: Thermo-piezoelectric Scatterers. Fluids, 6(3). doi:10.3390/fluids6030101
• Sánchez, N., Sánchez-Vizuet, T., & Solano, M. E. (2021). A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems. Numerische Mathematik. doi:10.1007/s00211-021-01221-8
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  We present a priori and a posteriori error analysis of a high orderhybridizable discontinuous Galerkin (HDG) method applied to a semi-linearelliptic 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 non-linear source term and the distance between the boundaries of thepolygonal subdomain $\Omega_h$ and the true domain $\Omega$. Moreover, a localnon-linear post-processing 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: ]
• Sanchez-Vizuet, T., & Hsiao, G. C. (2020). Time-Domain Boundary Integral Methods in Linear Thermoelasticity. SIAM Journal on Mathematical Analysis.
• Sanchez-Vizuet, 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., Sanchez-vizuet, T., Sayas, F., & Weinacht, R. J. (2019). A time-dependent wave-thermoelastic solid interaction. Ima Journal of Numerical Analysis, 39(2), 924-956. doi:10.1093/imanum/dry016
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  This paper presents a combined field and boundary integral equation method for solving time-dependent 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 time-dependent fluid-structure 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 time-dependent boundary integral equations. Using the Laplace transform in terms of time-domain data existence and uniqueness results are established. Galerkin semi-discretization 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.
• Sanchez-Vizuet, T., & Solano, M. E. (2019). A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation. Computer Physics Communications.
• Brown, T. S., Sanchez-vizuet, 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), 423-455. doi:10.1051/m2an/2017045
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  We 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. Well-posedness 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.
• Sanchez-Vizuet, 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), 764-778. doi:10.1090/noti1706
• Sanchez-Vizuet, T., Sayas, F., Hsiao, G. C., & Weinacht, R. (2018). A Time-Dependent Wave-Thermoelastic Solid Interaction. IMA Journal of Numerical Analysis, 39(2), 924–956.
• Sanchez-vizuet, 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:10.1088/1361-6587/aa963a
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  We study the approximation and stability properties of a recently popularized discretization strategy for the speed variable in kinetic equations, based on pseudo-spectral collocation on a grid defined by the zeros of a non-standard family of orthogonal polynomials called Maxwell polynomials. Taking a one-dimensional 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 high-dimensional gyrokinetic simulations. However, we also show that Maxwell based schemes are subject to a non-modal 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., Sanchez-vizuet, T., & Sayas, F. (2017). Boundary and coupled boundary–finite element methods for transient wave–structure interaction. Ima Journal of Numerical Analysis, 37(1), 237-265. doi:10.1093/imanum/drw009
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  We propose time-domain 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 BDF2-CQ and numerical experiments are provided for both BDF2 and Trapezoidal Rule CQ showing second order behavior for the latter as well.
• Sanchez-Vizuet, T., & Cerfon, A. (2017). Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion. Plasma Physics and Cotrolled Fusion.
• Sanchez-Vizuet, T., & Sayas, F. (2017). Symmetric Boundary-Finite Element Discretization of Time Dependent Acoustic Scattering by Elastic Obstacles with Piezoelectric Behavior. Journal of Scientific Computing.
• Sanchez-Vizuet, 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.
• Sanchez-Vizuet, 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.
• Sanchez-Vizuet, T., Sayas, F., & Hsiao, G. (2016). Boundary and coupled boundary–finite element methods for transient wave–structure interaction. IMA Journal of Numerical Analysis.
• Sanchez-Vizuet, T., Sayas, F., & Dominguez, V. (2015). A fully discrete Calderón calculus for the two-dimensional elastic wave equation. Computers & Mathematics with Applications.

Proceedings Publications

• Askham, T., Ball, J., Cerfon, A., Freidberg, J., Greenwald, M., Imbert-Gérard, L., Kim, E., Landreman, M., Lee, J., Majda, A., Malhotra, D., McFadden, G., O'Neil, M., Parra, F., Qi, D., Rachh, M., Ricketson, L., Sanchez-Vizuet, T., Segal, D., & Wilkening, J. (2022).

  High Performance Equilibrium Solvers for Integrated Magnetic Fusion Simulations

  . In DOE Technical report.

Presentations

• Sanchez-Vizuet, T. (2023, October). A high order solver for the Grad-Shafranov free boundary problem. 56th Annual meeting of the Mexican Mathematical Society. San Luis Potosi, Mexico: Mexican Mathematical Society.