Abstract : Many phenomena and problems in modern science, technology and engineering can be described by partial differential equations. For example, the Gross-Pitaevskii equation under a rotational frame, the nonlinear Klein-Gordon-Schr\"{o}dinger equations in the nonrelativistic limit regime, Poisson-Nernst-Planck systems, etc. It is significant to design efficient numerical methods to solve the above PDEs with numerical analysis and provide an intuitive view for physical phenomena. The main purpose of this mini-symposium is to discuss recent developments of the numerical methods for solving time-dependent PDEs.
[02681] Unconditionally MBP-preserving linear schemes for conservative Allen-Cahn equations
Format : Talk at Waseda University
Author(s) :
Jingwei Li (Lanzhou University)
Abstract : The maximum bound principle MBP, is an important property for semilinear parabolic equations, in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions and nonlinear operator preserves for all time a uniform pointwise bound in absolute value. It has been a challenging problem to design unconditionally MBP-preserving high-order accurate time-stepping schemes for these equations. Du Qiang et al have estiblished a unified analysical framework on the MBP preserving scheme for the semilinear parabolic equations which in this talk will be extended to conservative Allen-Cahn equation with the introduced Lagrange multiplier enforcing the mass conservation. Some sufficient conditions on the nonlinear potentials will be given under which the MBP holds and then the stabilized exponential time differencing scheme is proposed for time integration, which are linear schemes and unconditionally preserve the MBP in the time discrete level. Convergence of these schemes is analyzed as well as their energy stability. Various two and three dimensional numerical experiments are also carried out to validate the theoretical results and demonstrate the performance of the proposed schemes. These work are joint with Cai Yongyong, Feng Xinlong, Huang Qiumei, Jiang Kun, Ju Lili, Li Xiao, Lan Rihui et al.
[02717] Uniformly accurate nested Picard iterative integrators for the Klein-Gordon-Schr\"{o}dinger equation in the nonrelativistic regime
Format : Online Talk on Zoom
Author(s) :
xuanxuan zhou (beijing normal university)
Abstract : We establish a class of uniformly accurate nested Picard iterative integrator (NPI) Fourier pseudospectral methods for the nonlinear Klein-Gordon-Schr\"{o}dinger equation (KGS) in the nonrelativistic regime, involving a dimensionless parameter $\varepsilon\ll1$ inversely proportional to the speed of light. Actually, the solution propagates waves in time with $O(\varepsilon^2)$ wavelength when $0<\varepsilon\ll1$, which brings significant difficulty in designing accurate and efficient numerical schemes. The NPI method is designed by separating the oscillatory part from the non-oscillatory part, and integrating the former exactly. Based on the Picard iteration, the NPI method can be applied to derive arbitrary higher-order methods in time with optimal and uniform accuracy (w.r.t. $\varepsilon\in(0,1]$), and the corresponding error estimates are rigorously established. In addition, the practical implementation of the second-order NPI method via Fourier pseupospectral discretization is clearly demonstrated, with extensions to the third order NPI. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the proposed schemes.
[02550] Structure-preserving scheme for the PNP equations
Format : Talk at Waseda University
Author(s) :
Fenghua Tong (Beijing Normal University)
Abstract : Poisson-Nernst-Planck system is a macroscopic model to describe the ion transport process. We propose a novel method to construct the positivity preserving and mass conservation scheme for the Poisson-Nernst-Planck equations. The method is based on the discrete $L^2_h$ or $H^1_h$ projection strategy in which the solution projected from the intermediate solution computed by semi-implicit scheme inherits the positivity preservation and mass conservation with negligible additional computational cost resulting from the nonlinear algebraic equation.
[02553] Numerical simulation of rotational nonlinear Schrodinger equations with attractive interactions
Format : Talk at Waseda University
Author(s) :
Yong Wu (Beijing Normal University)
Abstract : We consider the focusing Schr$\ddot{o}$dinger equation with rotation and numerically simulate the ground state and dynamic properties. We take the gradient flow with Lagrange multiplier (GFLM) method to compute the ground state and time splitting pseudospectral method to simulate dynamics. We numerically verify the nonexistence of vortices in harmonic symmetry potential and analytically derive that the symmetric state energy (m=0) is always lower than the central vortex state energy (m=1) when the attractive interaction is sufficiently small. For dynamics properties, we mainly simulated the condensate widths, a stationary state with a shifted center and stability of central vortex states and found that the conclusion is completely consistent with the repulsive interaction. Finally, we numerically simulate the global existence and finite time blow-up of solution in mass-supercritical case.
[02622] IMPROVED UNIFORM ERROR BOUNDS OF THE TIME-SPLITTING HERMITE SPECTRAL METHODS FOR THE LONG-TIME GROSS PITAEVSKII EQUATION WITH WEAK NONLINEARITY
Format : Online Talk on Zoom
Author(s) :
Zhongyang Liu (Beijing Normal University)
Abstract : The aim of this research is to carry out a improved uniform error bounds for the Strang splitting Hermite pseudospectral methods for the long-time dynamics of the time-dependent Gross–Pitaevskii equation (GPE) with weak nonlinearity, while the nonlinearity strength is characterized by $\epsilon^2$ with a dimensionless parameter $\epsilon\in (0,1]$, for the long time dynamics up to the time at $O(\epsilon^{-2})$. We derive a improved uniform $H_A^1$ error bounds for full discretizations of the one-dimensional GPE by the Strang splitting Hermite pseudospectral method as $O(N^{\frac{2}{3}-\frac{m}{2}}+\epsilon^2\tau^2)$ up to the time at $O(1/\epsilon^2)$. The error bounds are uniformly accurate up to the time at $O(\epsilon^{-2})$ and uniformly valid for $\epsilon.$
