TY - JOUR
T1 - On a Parabolic Sine-Gordon Model
AU - Cheng , Xinyu
AU - Li , Dong
AU - Quan , Chaoyu
AU - Yang , Wen
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 1068
EP - 1084
PY - 2021
DA - 2021/09
SN - 14
DO - http://doi.org/10.4208/nmtma.OA-2021-0040
UR - https://global-sci.org/intro/article_detail/nmtma/19530.html
KW - Sine-Gordon equation, backward differentiation formula, implicit-explicit scheme.
AB - <p style="text-align: justify;">We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform
control of the solution. In the one-dimensional case we classify all bounded steady
states and exhibit some explicit solutions. For the numerical discretization we employ first order IMEX, and second order BDF2 discretization without any additional
stabilization term. We rigorously prove the energy stability of the numerical schemes
under nearly sharp and quite mild time step constraints. We demonstrate the striking
similarity of the parabolic sine-Gordon model with the standard Allen-Cahn equations with double well potentials.</p>