NumPI April 2021

numpi@lists.dm.unipi.it

1 participants
2 discussions

Seminar on 23/04 (Philipp Birken)
by Leonardo Robol 23 Apr '21

23 Apr '21

Speaker: Philipp Birken Affiliation: Lund University Time: Friday, 23/04/2021, 16:00 Title: Conservative iterative solvers in computational fluid dynamics The governing equations in computational fluid dynamics such as the Navier-Stokes- or Euler equations are conservation laws. Finite volume methods are designed to respect this and the theorem of Lax-Wendroff underscores the importance of it. It roughly states that for a nonlinear (!) scalar conservation law in 1D , if the numerical method with explicit Euler time integration is consistent and (locally) conservative, then in case of convergence, the numerical method converges to a weak solution. When using implicit time integration, the widespread believe in the community is that conservation is lost. This is however, not necessarily due to the time integration, but due to the use of iterative solvers. We first present a catalogue of iterative solvers that preserve the weaker property of global conservation to identify candidates of solvers that preserve local conservation as used in the Lax-Wendroff theorem. We then proceed to prove an extension of the Lax-Wendroff theorem for the situation that we perform a fixed number of steps of a so called pseudo time iteration per time step. It turns out that in this case, the numerical method converges to a weak solution of the conservation law with a modified propagation speed. This can be exploited to improve performance of the iterative method. https://www.dm.unipi.it/webnew/it/seminari/conservative-iterative-solvers-c… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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Seminar on 09/04 (Jie Meng)
by Leonardo Robol 09 Apr '21

09 Apr '21

Speaker: Jie Meng Affiliation: Università di Pisa Time: Friday, 09/04/2021, 16:00 Title: Geometric means of quasi-Toeplitz matrices We study means of geometric type of quasi-Toeplitz matrices, that are semi-infinite matrices A = (a_{i,j}) i,j=1,2,... of the form A = T(a) + E, where E represents a compact operator, and T(a) is a semi-infinite Toeplitz matrix associated with the function a, with Fourier series \sum_{l} a_l e^{ilt} , in the sense that (T(a))_{i,j} = a_{j-i}. If a is real valued and essentially bounded, then these matrices represent bounded self-adjoint operators on l^2. We consider the case where a is a continuous function, where quasi-Toeplitz matrices coincide with a classical Toeplitz algebra, and the case where a is in the Wiener algebra, that is, has absolutely convergent Fourier series. We prove that if a_1, ... , a_p are continuous and positive functions, or are in the Wiener algebra with some further conditions, then means of geometric type, such as the ALM, the NBMP and the Karcher mean of quasi-Toeplitz positive definite matrices associated with a_1, ..., a_p , are quasi-Toeplitz matrices associated with the geometric mean (a_1 ... a_p)^{1/p}, which differ only by the compact correction. We show by numerical tests that these operator means can be practically approximated. https://www.dm.unipi.it/webnew/it/seminari/geometric-means-quasi-toeplitz-m… Meeting link: https://hausdorff.dm.unipi.it/b/leo-xik-xu4

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NumPI April 2021