Pengde Wang | Searcharxiv

arXiv2024

The stochastic fractional nonlinear Schr\"odinger equations in $H^{\alpha}$ and structure-preserving algorithm

In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schr\"odinger equation with radially symmetric initial data in a suitable energy space $H^{\alpha}$. We then show that the stochastic fractional nonlinear Schr\"odinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity. Finally, we develop a stochastic midpoint scheme for the stochastic fractional nonlinear Schr\"odinger equation from the perspective of symplectic geometry. It is proved that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. A numerical example is conducted to validate the efficiency of the theory.

math.NA↗

arXiv2016

An implicit midpoint difference scheme for the fractional Ginzburg-Landau equation

This paper proposes and analyzes an efficient difference scheme for the nonlinear complex Ginzburg-Landau equation involving fractional Laplacian. The scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted Gr\"unwald difference operator for the spatial fractional Laplacian. By virtue of a careful analysis of the difference operator, some useful inequalities with respect to suitable fractional Sobolev norms are established. Then the numerical solution is shown to be bounded, and convergent in the $l^2_h$ norm with the optimal order $O(\tau^2+h^2)$ with time step $\tau$ and mesh size $h$. The a priori bound as well as the convergence order hold unconditionally, in the sense that no restriction on the time step $\tau$ in terms of the mesh size $h$ needs to be assumed. Numerical tests are performed to validate the theoretical results and effectiveness of the scheme.

math.NA↗