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
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
