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SUMMARY:Lie-Poisson methods for isospectral flows and their application to
  long-time simulation of spherical ideal hydrodynamics - Milo Viviani (Cha
 lmers University of Technology)
DTSTART:20190828T140000Z
DTEND:20190828T150000Z
UID:TALK129175@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The theory of isospectral flows comprises a large class of con
 tinuous dynamical systems\, particularly&nbsp\;integrable systems and Lie&
 ndash\;Poisson&nbsp\;systems. Their discretization is a classical problem 
 in numerical&nbsp\;analysis. Preserving the spectra in the discrete flow r
 equires the conservation of&nbsp\;high order polynomials\,&nbsp\;which is 
 hard to come by. Existing methods achieving this are complicated and usual
 ly fail to preserve&nbsp\;the underlying Lie-Poisson structure. Here we pr
 esent a class of numerical methods of arbitrary order&nbsp\;for Hamiltonia
 n and non-Hamiltonian isospectral flows\, which&nbsp\;preserve both the sp
 ectra and the Lie-Poisson structure. The methods are surprisingly simple\,
  and avoid the use of constraints or exponential&nbsp\;maps. Furthermore\,
  due to preservation of&nbsp\;the Lie&ndash\;Poisson structure\, they exhi
 bit near conservation&nbsp\;of the Hamiltonian function. As an illustratio
 n\, we apply the methods to long-time&nbsp\;simulation of the&nbsp\;Euler 
 equations on a sphere. Our findings suggest that our structure-preserving 
 algorithms\, on the&nbsp\;one hand\, perform at least as&nbsp\;well as oth
 er popular methods (i.e. CLAM) without adding spurious&nbsp\;hyperviscosit
 y terms\, on the other hand\, show that the conservation of the&nbsp\;Casi
 mir functions can be&nbsp\;actually used to predict the final state of the
  fluid<br><br><br><br><br>
LOCATION:Seminar Room 2\, Newton Institute
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