| Date: | September 26, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Xiangxiong Zhang, Purdue University |
| Title: | Positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations |
| Abstract: | For gas dynamics equations such as compressible Euler and Navier-Stokes equations, preserving the positivity of density and pressure without losing conservation is crucial to stabilize the numerical computation. The L1-stability of mass and energy can be achieved by enforcing the positivity of density and pressure during the time evolution. However, high order schemes such as DG methods do not preserve the positivity. It is difficult to enforce the positivity without destroying the high order accuracy and the local conservation in an efficient manner for time-dependent gas dynamics equations. For compressible Euler equations, a weak positivity property holds for any high order finite volume type schemes including DG methods, which was used to design a simple positivity-preserving limiter for high order DG schemes by Zhang and Shu in 2010. Generalizations to compressible Navier-Stokes equations are however nontrivial. We show that weak positivity property still holds for DG method solving compressible Navier-Stokes equations if a proper penalty term is added in the scheme. This allows us to obtain the first high order positivity-preserving schemes for compressible Navier-Stokes equations. |
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| Date: | October 10, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Ricardo Alonso |
| Title: | Convergence and error estimates for a conservative spectral method for the homogeneous Boltzmann equation. |
| Abstract: | Abstract: We develop error estimates for a semi-discrete conservative spectral method for the Boltzmann equation in L2 and Sobolev spaces. We prove that the conservation laws are, in some sense, more important than positivity of the solution in order to have accurate simulations for large times. Convergence of the numerical approximation to the Maxwellian equilibrium is also proved. |
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| Date: | October 29, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 624 |
| Speaker: | Agnieszka Miedlar, The University of Kansas |
| Title: | The NLFEAST Algorithm for Large-Scale Nonlinear Eigenvalue Problems |
| Abstract: | Eigenvalue problems in which the coefficient matrices depend nonlinearly on the eigenvalues arise in a variety of applications in science and engineering, e.g., dynamic analysis of structures or computational nanoelectronics, to mention just a few. This talk will discuss how the Cauchy integral-based approaches offer an attractive framework for using "spectrum slicing" to develop highly efficient and flexible techniques for solving large-scale nonlinear eigenvalue problems. We will first introduce the nonlinear counterpart of the well-established linear FEAST algorithm. Like its linear predecessor, the nonlinear FEAST (NLFEAST) algorithm can be used to solve nonlinear eigenvalue problems for the eigenpairs corresponding to eigenvalues that lie in a user-specified region in the complex plane, thereby allowing for the calculation of large number of eigenpairs in parallel. To develop a nonlinear FEAST algorithm that enables the iterative refinement of a subspace of a fixed dimension, we propose to use a modified form of the contour integral resulting from the relationship between the NLFEAST and the residual inverse iteration by Neumaier for the nonlinear eigenvalue problems. Finally, we will use several physically-motivated examples to illustrate the method. This is a joint work with B. Gavin, E. Polizzi and Y. Saad.
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| Date: | October 31, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Juan-Pablo Borthagaray, University of Maryland |
| Title: | Finite elements for fractional diffusion: towards nonlinear problems |
| Abstract: | In this talk we consider problems involving the integral fractional Laplacian on bounded domains. The first part is devoted to analysis of linear problems; we discuss regularity of solutions, analyze direct finite element implementations and derive convergence rates. Afterwards we discuss two nonlinear problems: the fractional obstacle problem and the computation of nonlocal minimal surfaces. The integral fractional Laplacian is a nonlocal operator given by a singular integral (defined in the principal value sense). Therefore, suitable quadrature is required to handle the singularity of the kernel. Nonlocality originates additional difficulties, such as the need to cope with integration on unbounded domains and full stiffness matrices. Independently of the smoothness of the domain and the data, solutions to the problems under consideration possess a limited Sobolev regularity. In order to enhance the order of convergence of the finite element approximations, we introduce suitably defined weighted Sobolev spaces. This, in turn, leads to the consideration of discrete solutions on graded meshes and permits to obtain optimal convergence rates in two-dimensional domains. |
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| Date: | November 7, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Remi Abgrall, Universtät Zürich |
| Title: | Some remarks about the conservation properties of numerical schemes for hyperbolic problems. |
| Abstract: | Since the celebrated theorem by Lax and Wendroff (1960), The numerical discretisation of hyperbolic problems is done using numerical flux. This theorem, provided some stability estimates hold true, guaranty that any converging sub-sequence of numerical solutions will converge to a weak solution of the original problem. There is a straightforward generalisation to entropy inequalities. Since then, the game is to define 'good' numerical flux that are robust enough, or preserve some invariance properties, for example.
Indeed, not any 'good' numerical scheme do not write in term of flux, at least at first glance. An example is given by stabilized finite element schemes, like SUPG with artificial dissipation to control numerical oscillations in discontinuity. However, it can be shown, at least in the scalar case and equiped with suitable shock capturing term, to converge towards the entropy solution of the problem.
In this talk, I will show a slightly more general version of the discrete conservation property. This enable to show that any reasonable scheme is equivalent to a finite volume scheme, i.e. it is possible to construct explicitely numerical flux so that the algebraic structure of the method is kept the same. The same approach enables also to modify any existing scheme so that the modified scheme will also satisfy, in addition, one given conservation relation. Example of entropy conservative scheme will be shown.
This is a joint work with S. Tokareva (now in Los Alamos) and P. Bacigaluppi (University of Zurich) |
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| Date: | November 28, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Per-Olof Persson, UC Berkely |
| Title: | High-Order Discontinuous Galerkin Methods for Fluid and Solid Mechanics |
| Abstract: | It is widely believed that high-order accurate numerical methods, for example discontinuous Galerkin (DG) methods, will eventually replace the traditional low-order methods in the solution of many problems, including fluid flow, solid dynamics, and wave propagation. The talk will give an overview of this field, including the theoretical background of the numerical schemes, the efficient implementation of the methods, and examples of real-world applications. Topics include high-order compact and sparse numerical schemes, high-quality unstructured curved mesh generation, scalable preconditioners for parallel iterative solvers, fully discrete adjoint methods for PDE-constrained optimization, and implicit-explicit schemes for the partitioning of coupled fluid-structure interaction problems. The methods will be demonstrated on some important practical problems, including the inverse design of energetically optimal flapping wings and large eddy simulation (LES) of wind turbines. |
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| Date: | November 29, 2018 |
| Time: | 1:00pm |
| Location: | BLOC 220 |
| Speaker: | Shira Faigenbaum, Tel Aviv University |
| Title: | Repairing and Denoising Scattered Data for the Reconstruction of Manifolds Embedded in High Dimensions |
| Abstract: | High dimensional data is increasingly available in many fields, and the problem of extracting valuable information from such data is of primal interest. A common assumption is that high dimensional data is an embedding of a low dimensional manifold. Often, the data suffers from presence of noise, outliers, and non-uniform sampling (which may result in 'holes' in the manifold). Standard approximation tools fail to address those problems - even in low dimensions. In our research, we propose to reconstruct the manifold's geometry by extending the Locally Optimal Projection operator (LOP) algorithm to the high dimensional case. Additionally, we utilize our framework to address other challenges rising in high dimensional data: a) calculation of k-multivariate L1-medians; b) smooth manifold repairing; c) up/down data sampling. We will demonstrate the effectiveness of our framework by considering noisy data from manifolds of 2-10 dimensions embedded in R^60. |
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