TY - JOUR
T1 - Invariants-Preserving Du Fort-Frankel Schemes and Their Analyses for Nonlinear Schrödinger Equations with Wave Operator
AU - Deng , Dingwen
AU - Li , Zhijun
JO - Journal of Computational Mathematics
VL - 3
SP - 814
EP - 850
PY - 2024
DA - 2024/04
SN - 42
DO - http://doi.org/10.4208/jcm.2211-m2021-0293
UR - https://global-sci.org/intro/article_detail/jcm/23037.html
KW - Nonlinear Schrödinger equations with wave operator, Du Fort-Frankel finite difference methods, Discrete energy and mass conservative laws, Numerical convergence.
AB - <p style="text-align: justify;">Du Fort-Frankel finite difference method (FDM) was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953. It is
an explicit and unconditionally von Neumann stable scheme. However, there has been no
research work on numerical solutions of nonlinear Schrödinger equations with wave operator
by using Du Fort-Frankel-type finite difference methods (FDMs). In this study, a class of
invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional
(1D) and two-dimensional (2D) nonlinear Schrödinger equations with wave operator. By
using the discrete energy method, it is shown that their solutions possess the discrete energy
and mass conservative laws, and conditionally converge to exact solutions with an order of $\mathcal{O}(τ^2+h^2_x+(τ/h_x)^2)$ for 1D problem and an order of $\mathcal{O}(τ^2+h^2_x+h^2_y+(τ/h_x)^2+(τ/h_y)^2)$ for 2D problem in $H^1$-norm. Here, $τ$ denotes time-step size, while, $h_x$ and $h_y$ represent
spatial meshsizes in $x$- and $y$-directions, respectively. Then, by introducing a stabilized
term, a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised.
They not only preserve the discrete energies and masses, but also own much better stability
than original schemes. Finally, numerical results demonstrate the theoretical analyses.</p>