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 -

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.