ABSTRACTS | Recent trends for the numerical solution of Partial Differential Equations

Abstract
In this short course, we present a systematic and constructive derivation of hybridized discontinuous Galerkin (HDG) methods using an upwind framework. The key behind our construction is the Godunov method and the Rankine-Hugoniot condition. We will show in details the construction of an upwind HDG framework and its application to constructively derive HDG methods for a large class of partial differential equations (PDEs) including elliptic, parabolic, hyperbolic, and mixed-types PDEs. The participants will learn how to apply this framework to derive HDG methods for convection-diffusion-reactions equations (and its subsets), Maxwell
equations, and linearized shallow water equation. The well-posedness of the HDG framework will be discussed and the well-posedness analysis of the HDG formulation for the convection-diffusion-reaction will be carried out in details. The participants will learn a step-by-step convergence analysis of the HDG method for the linearized shallow water equations. We will discuss the extension to nonlinear systems and active research topics.

At the end of the short course, participants will, without wary, be able to construct HDG methods for their applications of interest. They will learn how HDG is implemented in practice and accompanied HDG Matlab codes will be provided at end of the short course.

On the numerical solution of non-smooth direct and inverse problems
Roland Glowinski
University of Houston, Houston, Texas

Abstract

CONFERENCIAS

A Fast and Accurate Vortex Method on the GPU
Julián T. Becerra Sagredo
Mathematics Department, UAM-Iztapalapa

Abstract
The vortex method is a numerical scheme where particles with vorticity are used for the simulation of incompressible flows with turbulence. In order to impose the incompressibility condition, the system of equations is closed with a Poisson equation for the stream function. For its implementation in GPUs (graphic processing units), we have used a kind of multi-grid (GSGML) for solution of the Poisson equation and a moment-preserving semi-Lagrangian scheme for the advection of the vortices. All the schemes have been developed specifically to be implemented entirely on the GPU. We show comparisons of the results of the method with many other schemes for the case of under-resolved turbulent flows. The method is capable of improving the results of previous schemes and, to our best knowledge, is the first full implementation of a vortex method on the GPU.

A geometric description of discrete exterior calculus in 2D
Rafael Herrera
Mathematics Department, CIMAT

Abstract
We will present the theory of Discrete Exterior Calculus (DEC) in 2D using only Vector Calculus and Matrix Algebra, and a formulation of DEC for general triangulations, thus removing the restriction of well-centered meshes.

This formulation is tested numerically on the Poisson equation and the solutions are compared against those found using the Finite Element Method with linear elements (FEML).

On the computation of Fourier Integral Operators in problems connected to wave propagation PDEs.
J. Héctor Morales Bárcenas
Mathematics Department, UAM-Iztapalapa

Abstract
In this talk, we present numerical implementations of forward and inverse maps in terms of Fourier Integral Operators (FIOs) of wave propagation and tomography. The idea to represent PDEs solutions in terms of FIOs is that we are close to interpret them like a generalization of Fourier transforms (a well stablished methodology to solve linear and constant coefficient PDEs). Instead, FIOs represent solutions of non-constant coefficient PDEs, however these objects may be highly oscillatory and multidimensional integrals, that represent by it self a numerical challenge. We will exemplify FIOs in action in tomography.

Reliability assessment of a surrogate model of wave propagation in functionally graded materials
Marcos Capistrán
CIMAT

Abstract
We assess the reliability of explicit formulas introduced by Pham et al. to compute the propagation of one-component waves through a stack of homogeneous layers, inserted between two half-planes, as a surrogate model of wave propagation in functionally graded materials (FGM). We have considered the continuous nodal Galerkin method (CG) as a reference to simulate from the continuous problem. Of note, the continuos nodal Galerkin method has spectral convergence, and its computational cost is cubic with respect to the number of degrees of freedom in the discretization. Consequently, we have compared computationally the output of the surrogate model and the continuous nodal Galerkin (CG) discretization of the continuous problem through sensitivity analysis; and secondly, we have used the surrogate model and the Bayesian inferential framework to estimate the FGM shear modulus given measurements of the amplitude of the reflected wave. Our motivation is that Pham formulas are asymptotically correct up to second order with respect to the number of layers for a wide range of frequencies, while the computational cost of computing the amplitude of elastic waves reflected by the FGM increases linearly. We have found that if we know the structure of noise in the data and the signal to noise ratio, then we can reliably infer the FGM shear modulus using the surrogate model and parallel tempering on the arising posterior distribution.

An inverse problem for the shallow water equations
Gerardo Hernández-Dueñas
Instituto de Matemáticas, UNAM

Abstract
Given a velocity measured at distinct points in space and time, an inverse problem is formulated to estimate the bottom topography in a shallow water flow. In this talk, we will present the least squares functional of the PDE-constrained minimization problem, the adjoint equations and the Roe-type upwind numerical schemes to solve the direct and inverse problems. This is joint work with Miguel Ángel Moreles and Pedro González-Casanova.

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