BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Trung Hau Hoang\, Allar Liivlaid DTSTART:20261007T070000Z DTEND:20261007T080000Z DTSTAMP:20261024T105615Z UID:MathMAC/64 DESCRIPTION:Title: Exponential Integrators: From Implementations to Order Reduction\, Simula tion of incompressible particle-based fluid flow inspired by optimal trans port\nby Trung Hau Hoang\, Allar Liivlaid as part of Modelling of mate rials - theory\, model reduction and efficient numerical methods (UNCE Mat hMAC)\n\n\nAbstract\nExponential Integrators: From Implementations to O rder Reduction
\nExponential integrators are a class of numerical m ethods designed for the efficient integration of evolution equations with stiff linear parts. By treating the linear operator through matrix exponen tials\, these methods provide an attractive approach to problems arising f rom the spatial discretization of partial differential equations. We provi de an overview of exponential integrators\, with an emphasis on efficient implementation and applications to evolution equations involving unbounded operators. We introduce the basic principles of exponential Runge--Kutta methods\, together with a brief overview of several approaches for computi ng matrix exponentials and $\\varphi$-function actions\, including Krylov subspace\, Leja interpolation\, and Chebyshev approximation methods. In th is context\, we compare these approximation methods with explicit Runge--K utta methods of orders two and four for computing the action of the expone ntial and related matrix functions. A performance model is established by accounting for the cost of the different operations required to implement the considered algorithms. The results show that\, depending on the specif ic setting\, exponential integrators either outperform or perform similarl y to explicit Runge--Kutta schemes. We also discuss order reduction in exp onential Runge--Kutta methods for problems involving unbounded and non-com muting operators. The mechanisms leading to order reduction are discussed. The analysis shows that first- and second-order methods retain their expe cted order under mild regularity conditions on the initial data $u_0$\, wh ereas third- and fourth-order methods can exhibit order reduction. In part icular\, for third-order methods\, the analysis identifies schemes for whi ch order reduction can be essentially eliminated under suitable regularity assumptions. Numerical experiments illustrating the theoretical results a re provided. The talk concludes with perspectives on the analysis and effi cient implementation of exponential integrators for stiff evolution equati ons and partial differential equations.

\n\nSimulation of incompressi ble particle-based fluid flow inspired by optimal transport
\nThe s imulation of particle-based fluid flow is usually compressible due to non- constant density\, finite speed of sound from interaction effects\, and la ck of a global pressure field. By combining results from optimal transport theory\, especially a globally optimal assignment problem solver with a f ormulation corresponding to an incompressible particle-based fluid flow\, the incompressible Euler flow was first approximated in a new computationa l formulation. After inclusion of a viscosity step\, known qualitative sol utions for the Navier-Stokes equation can be demonstrated\, such as for a Karman vortex street. These showed the initial suitability of the method f or solving inviscid and viscous flow problems and hopefully\, after succes sful validation\, the introduced solver will be efficient for such problem s as homogenization flows and mixtures of different density fluids.\n LOCATION:https://researchseminars.org/talk/MathMAC/64/ END:VEVENT END:VCALENDAR