Events
Department of Mathematics and Statistics
Texas Tech University
We study the precise asymptotic behavior of a non-trivial solution that converges to zero, as time tends to infinity, of dissipative systems of nonlinear ordinary differential equations. The nonlinear term of the equations may not possess a Taylor series expansion about the origin. This absence technically cripples previous proofs in establishing an asymptotic expansion, as an infinite series, for such a decaying solution. In the current paper, we overcome this limitation and obtain an infinite series asymptotic expansion, as time goes to infinity. This series expansion provides large time approximations for the solution with the errors decaying exponentially at any given rates. The main idea is to shift the center of the Taylor expansions for the nonlinear term to a non-zero point. Such a point turns out to come from the non-trivial asymptotic behavior of the solution, which we prove by a new and simple method. Our result applies to different classes of non-linear equations that have not been dealt with previously. This is joint work with Dat Cao (Minnesota State University, Mankato) and Thinh Kieu (University of North Georgia, Gainesville Campus).This week's Biomath seminar details available at this pdf
 | Wednesday
Feb. 24
|
| Algebra and Number Theory
No Seminar
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This week's PDGMP seminar details available at this pdf
Please virtually attend Dr. Aulisa's presenation at 3 PM on Wednesday the 24th at this zoom link.
We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a space-time region partitioned by time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into account by a new flux term, and can be integrated by method-of-line Runge-Kutta (RK) methods. The ELDG framework is designed as a generalization of the semi-Lagrangian (SL) DG method and classical Eulerian RK DG method for linear advection problems. It takes advantages of both formulations. In the EL DG framework, characteristics are approximated by a linear function in time, thus shapes of upstream cells are quadrilaterals in general two-dimensional problems. No quadratic-curved quadrilaterals are needed to design higher than second order schemes as in the SL DG scheme. On the other hand, the time step constraint from a classical Eulerian RK DG method is greatly mitigated, as it is evident from our theoretical and numerical investigations. Connection of the proposed EL DG method with the arbitrary Lagrangian-Eulerian (ALE) DG is observed. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.
Please virtually attend this week's Applied Math seminar via this zoom link Wednesday the 24th at 4 PM.