BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Bangwei She (Inst. of Mathematics\, Czech Academy of Sciences) DTSTART:20220321T144000Z DTEND:20220321T161000Z DTSTAMP:20260405T174649Z UID:NSCM/68 DESCRIPTION:Title: On convergence of numerical solutions for the compressible MHD system\nb y Bangwei She (Inst. of Mathematics\, Czech Academy of Sciences) as part o f Nečas Seminar on Continuum Mechanics\n\n\nAbstract\nWe study a general convergence theory for the analysis of numerical\n solutions to a magnetoh ydrodynamic system describing the time\n evolution of compressible\, visco us\, electrically conducting fluids.\n \n First\, we introduce the concept of consistent approximation mimicking\n the density positivity\, energy s tability\, and consistency of a\n suitable numerical approximation. Furthe r\, we introduce the concept of\n dissipative weak solution\, which can be obtained as the weak limit of\n the consistent approximation. Here\, by “dissipative" we mean that\n the energy inequality contains energy defec ts that control the\n oscillations in the momentum equation.\n \n Next\, b y using the relative energy functional\, we prove the\n dissipative weak-s trong uniqueness principle\, meaning that a\n dissipative weak solution co incides with a classical solution of the\n same problem as long as the lat ter exists. This indicates that a\n consistent approximation converges unc onditionally to the classical\n solution. As a summary\, we built a nonlin ear variant of the Lax\n equivalence theory for the compressible MHD syste m.\n \n Finally\, to show the application of the convergence theory\, we p ropose\n two numerical methods. We show that the numerical solutions prese rve\n the positivity of density and stability of the total energy. Then by \n using the a priori estimates derived from the energy estimates we\n pro ve that the numerical methods are consistent. Consequently\, our\n numeric al methods belong to the class of consistent approximation.\n Applying the prebuilt convergence theory\, we conclude that the\n solutions of our num erical methods converge to i) the dissipative\n weak solution\; ii) the c lassical solution as long as the classical\n solution exists. As a byprodu ct of the first convergence\, we prove the\n global in-time existence of t he dissipative weak solution.\n LOCATION:https://researchseminars.org/talk/NSCM/68/ END:VEVENT END:VCALENDAR