Archive : Numerical Analysis and PDE

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Dr. Jingmei Qiu, University of Delaware150Dr. Jingmei Qiu, University of Delaware<p>​Title: A Conservative Low Rank Tensor Representation of Nonlinear Vlasov-Poisson and Vlasov-Maxwell Solutions</p><p><br>Abstract: We propose a low-rank tensor approach to approximate nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation is tensor friendly, based on which we propose to dynamically and adaptively build up low-rank solution basis by adding new basis functions from discretization of the PDE, and removing basis from an SVD-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization and a second order strong stability preserving multi-step time discretization. We apply the same procedure to evolve the dynamics of the flow map in a low-rank fashion, which proves to be advantageous when the flow map enjoys the low rank structure, while the solution suffers from high rank or displays filamentation structures. Local conservation properties are built in by a set of special projections. Hierarchical Tucker decomposition is adopted for high dimensional problems. An extensive set of linear and nonlinear Vlasov test examples are performed to show the high order spatial and temporal convergence of the algorithm with mesh refinement up to SVD-type truncation, the significant computational savings of the proposed low-rank approach especially for high dimensional problems, the improved performance of the flow map approach for solutions with filamentations.<br></p>12/3/2021 4:00:00 PM12/3/2021 5:00:00 PM
Dr. Yangwen Zhang, Carnegie Mellon University145Dr. Yangwen Zhang, Carnegie Mellon UniversityZoom<p>​<span class="wrap-text"><strong>Title:</strong> Sharp $L^\infty$ estimates of HDG methods for Poisson equation II: 3D<br><strong>Abstract:</strong> In [SIAM J. Numer. Anal., 59 (2), 720-745], we proved quasi-optimal $L^\infty$ estimates (up to logarithmic factors) for the solution of Poisson's equation by a hybridizable discontinuous Galerkin (HDG) method. However, the estimates \emph{only} work in 2D. In this paper, we use the approach in [Numer. Math., 131 (2015), pp. 771–822] and obtain \emph{sharp} (without logarithmic factors) $L^\infty$ estimates for the HDG method in both 2D and 3D. Numerical experiments are presented to confirm our theoretical result.</span><br></p>11/19/2021 4:00:00 PM11/19/2021 5:00:00 PM
Dr. Mark Gockenbach, University of Delaware147Dr. Mark Gockenbach, University of DelawareZoom<p><strong>​Title:</strong> Approximating the generalized singular value expansion<br><br><strong>Abstract: </strong> The generalized singular value expansion (GSVE) simultaneously diagonalizes a pair $(T,L)$ of operators defined on Hilbert spaces. Just as the singular value expansion (SVE) of a compact operator $T$ resolves many problems about $T$ (such as computing the pseudo-inverse, regularizing the inverse problem $Tx=y$, etc.), so does the GSVE for $(T,L)$ (computing the weighted pseudo-inverse of $T$, regularizing $Tx=y$ using $L$ as a regularization operator, etc.).<br><br>This talk will discuss the approximation of the GSVE of $(T,L)$ using Galerkin methods. The goal is to compare the errors in the approximate generalized singular vectors with the optimal errors relative to the chosen approximating subspaces; ideally, we would show that the computed solutions are asymptotically optimal. As we will see, we can get close to this goal but not achieve it.<br></p>11/12/2021 4:00:00 PM11/12/2021 5:00:00 PM
Dr. Xue Hong, University of Delaware149Dr. Xue Hong, University of Delaware<p><strong>​</strong><span class="wrap-text"><strong>Title:</strong> An Eulerian-Lagrangian discontinuous Galerkin method for wave equations<br><br><strong>Abstract:</strong> We propose an Eulerian-Lagrangian (EL) Runge-Kutta (RK) discontinuous Galerkin (DG) method for wave equations. The method is designed based on the ELDG method for transport problems proposed in [arXiv preprint arXiv: 2002.02930 (2020)], which tracks solution along approximations to characteristics in the DG framework, allowing extra large time stepping sizes with stability. The wave equation can be written as a first order hyperbolic system. Considering each characteristic family, a straight forward application of ELDG will be to project to the characteristic variables, evolve them on associated space-time regions, and project them back to the original variables. However, the mass conservation could not guaranteed in a general setting. In this paper, we formulated a mass conservative semi-discrete ELDG method by decomposing each variable into two parts, each of them associated with different characteristic families. As a result, four different quantities are evolved in EL fashion and recombined to update the solution. The fully discrete scheme is formulated by using method-of-lines RK methods, with intermediate RK solutions updated on the background mesh. Numerical results on 1D and 2D wave equations are presented to demonstrate the performance of the proposed ELDG method. These include the high order spatial and temporal accuracy, stability with extra large time stepping size, and mass conservative property.</span><br></p>10/29/2021 3:00:00 PM10/29/2021 4:00:00 PM

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