TY - JOUR
T1 - A Conservative Gradient Discretization Method for Parabolic Equations
AU - Zhou , Huifang
AU - Sheng , Zhiqiang
AU - Yuan , Guangwei
JO - Advances in Applied Mathematics and Mechanics
VL - 1
SP - 232
EP - 260
PY - 2020
DA - 2020/10
SN - 13
DO - http://doi.org/10.4208/aamm.OA-2020-0047
UR - https://global-sci.org/intro/article_detail/aamm/18349.html
KW - Gradient discretization method, mass conservation,  parabolic equations.
AB - <p style="text-align: justify;">In this paper, we propose a new conservative gradient discretization method (GDM) for one-dimensional parabolic partial differential equations (PDEs). We use the implicit Euler method for the temporal discretization and conservative gradient discretization method&nbsp; for spatial discretization. The method is based on a new cell-centered meshes, and it is locally conservative. It has smaller truncation error than the classical finite volume method on uniform meshes. We use the framework of the gradient discretization method to analyze the stability and convergence. The numerical experiments show that the new method has second-order convergence. Moreover, it is more accurate than the classical finite volume method in flux error, $L^2$ error and $L^{\infty}$ error.</p>