[00413] Numerical Methods for Dispersive PDEs and Applications
Session Date & Time :
00413 (1/2) : 4D (Aug.24, 15:30-17:10)
00413 (2/2) : 4E (Aug.24, 17:40-19:20)
Type : Proposal of Minisymposium
Abstract : Dispersive partial differential equations $\left({\rm PDEs}\right)$ play a fundamental role in many fields such as the nonlinear optics, water wave theory, quantum mechanics, etc. From the perspective of computational mathematics, it is significant to design efficient numerical methods to solve dispersive PDEs with in-depth numerical analysis and provide an intuitive view for physical phenomena. The proposed minisymposium invites experts in this field to review recent advances in numerical methods for dispersive PDEs and applications.
Christof Sparber (University of Illinois at Chicago)
Chunmei Su (Tsinghua University)
Chushan Wang (National University of Singapore)
Talks in Minisymposium :
[01752] Error estimates of numerical methods for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity
Author(s) :
Weizhu Bao (National University of Singapore)
Chushan Wang (National University of Singapore)
Abstract : We prove optimal error bounds of time-splitting methods and the exponential wave integrator for the nonlinear Schr\"{o}dinger equation (\text{(NLSE)}) with low regularity potential and nonlinearity, including purely bounded potential and locally Lipschitz nonlinearity. Arising from different physical applications, low regularity potential and nonlinearity are introduced into the NLSE such as the discontinuous potential or non-integer power nonlinearity, which make the error estimates of classical numerical methods very subtle and challenging.
[02732] Low regularity exponential-type integrators for the "good" Boussinesq equation
Author(s) :
Chunmei Su (Tsinghua University)
Hang Li (Tsinghua University )
Abstract : We introduce a series of semi-discrete low regularity exponential-type time integrators for the “good” Boussinesq equation. Compared to the existing numerical methods, the temporal convergence of ours can be achieved under weaker regularity assumptions on the exact solutions. The methods are constructed based on twisted variables and some harmonic analysis techniques in approximating the exponential integral. The methods are explicit and easy to be implemented efficiently when combined with pseudospectral method for spatial discretization.
[04017] Computational methods for stationary states of nonlinear Schrödinger/Gross-Pitaevskii equations
Author(s) :
Wei Liu (National University of Singapore)
Abstract : I will present some recent advances in the computation of stationary-state solutions to the nonlinear Schrödinger/Gross-Pitaevskii equations, primarily in the context of Bose-Einstein condensation. The (normalized) energy ground/excited states and action ground states will be mainly considered. Based on the analysis of variational characterizations and stabilities/instabilities for these stationary-state solutions, efficient and accurate numerical methods utilizing novel artificial dynamical flows and/or optimization techniques will be developed, with further extensions to challenging high-spin or fast-rotating models.
