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SUMMARY:E. I. Kaptsov (Suranaree University of Technology\, Thailand)
DTSTART:20230525T110000Z
DTEND:20230525T120000Z
DTSTAMP:20260423T040033Z
UID:mmandim/58
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/mmandim/58/"
 >Methods for constructing invariant conservative finite-difference schemes
  for hydrodynamic-type equations</a>\nby E. I. Kaptsov (Suranaree Universi
 ty of Technology\, Thailand) as part of Mathematical models and integratio
 n methods\n\n\nAbstract\nWhen choosing suitable finite-difference schemes 
 for equations of hydrodynamic type\, preference is given to various proper
 ties of schemes\, such as their monotonicity\, stability\, conservation of
  phase volumes\, etc. In the present report\, we focus on the criterion of
  invariance of schemes\, i.e. we consider finite-difference equations and 
 meshes that preserve the symmetries of the original differential equations
 .\n\nFor equations of the hydrodynamic type\, the construction of invarian
 t difference schemes is often significantly simplified if the equations ar
 e considered in Lagrange coordinates. In this case\, uniform orthogonal me
 shes can be used\, which retain their geometric structure under the action
  of group transformations inherited from the original equations. In additi
 on\, in Lagrangian  coordinates\, it is easier to find conservation laws b
 oth for differential equations and for the corresponding invariant differe
 nce schemes. In a number of cases\, it is possible to construct invariant 
 conservative schemes that possess difference analogues of all local conser
 vation laws of the original models.\n\nThe report is primarily devoted to 
 the practical aspects of designing schemes of the described type. For this
 \, a number of special techniques and methods have been developed. The mos
 t convenient is the finite-difference analogue of the direct method\, as w
 ell as the technique of constructing schemes based on approximations of co
 nservation laws.\nVarious equations of the theory of shallow water and one
 -dimensional equations of magnetohydrodynamics are considered as examples.
 \n\nReferences\n\n1. Dorodnitsyn V. A.\, Kaptsov E. I.\, Discrete shallow 
 water equations preserving symmetries and conservation laws. J. Math. Phys
 .\, 62(8):083508\, 2021.\n\n2. Kaptsov E. I.\, Dorodnitsyn V. A.\, Meleshk
 o S. V.\, Conservative invariant finite-difference schemes for the modifie
 d shallow water equations in Lagrangian coordinates. Stud. Appl. Math.\, 2
 022\; 149: 729–761.\n\n3. Dorodnitsyn V. A.\, Kaptsov E. I.\, and Melesh
 ko S. V.\, Symmetries\, conservation laws\, invariant solutions and differ
 ence schemes of the one-dimensional Green–Naghdi equations. J. Nonlinear
  Math. Phys.\, 28:90–107\, 2020.\n\n4. Cheviakov A. F.\, Dorodnitsyn V. 
 A.\, Kaptsov E. I.\, Invariant conservation law-preserving discretizations
  of linear and nonlinear wave equations\, J. Math. Phys.\, 61 (2020) P. 08
 1504.\n\n5. Dorodnitsyn V. A.\, Kaptsov E. I.\, Invariant finite-differenc
 e schemes for plane one-dimensional MHD flows that preserve conservation l
 aws. Mathematics\, 10(8):1250\, 2022.\n\n6. Kaptsov E. I.\, Dorodnitsyn V.
  A.\, Invariant conservative finite-difference schemes for the one-dimensi
 onal shallow water magnetohydrodynamics equations in Lagrangian coordinate
 s. Submitted. Preprint: <a href=https://arxiv.org/abs/2304.03488>https://a
 rxiv.org/abs/2304.03488</a>.\n\n7. Kaptsov E. I.\, Dorodnitsyn V. A.\, Mel
 eshko S. V.\, Invariant finite-difference schemes for cylindrical one-dime
 nsional MHD flows with conservation laws preservation. Submitted. Preprint
 : <a href=http://dx.doi.org/10.48550/arXiv.2302.05280>http://dx.doi.org/10
 .48550/arXiv.2302.05280</a>.\n
LOCATION:https://researchseminars.org/talk/mmandim/58/
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