TY - JOUR
T1 - Linearly Compact Difference Scheme for the Two-Dimensional Kuramoto-Tsuzuki Equation with the Neumann Boundary Condition
AU - Zhang , Qifeng
AU - Zhang , Lu
JO - Annals of Applied Mathematics
VL - 1
SP - 49
EP - 78
PY - 2023
DA - 2023/04
SN - 39
DO - http://doi.org/10.4208/aam.OA-2022-0015
UR - https://global-sci.org/intro/article_detail/aam/21633.html
KW - Kuramoto-Tsuzuki equation, compact difference scheme, pointwise error
estimate, stability, numerical simulation.
AB - <p style="text-align: justify;">In this paper, we analyze and test a high-order compact difference
scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki
equation under the Neumann boundary condition. A three-level average technique is utilized, thereby leading to a linearized difference scheme. The main
work lies in the pointwise error estimate in $H^2$-norm. The optimal fourth-order
convergence order is proved in combination of induction, the energy method
and the embedded inequality. Moreover, we establish the stability of the difference scheme with respect to the initial value under very mild condition, however, does not require any step ratio restriction. Extensive numerical examples
with/without exact solutions under diverse cases are implemented to validate
the theoretical results.</p>