TY - JOUR
T1 - Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations
AU - Liao , Hong-lin
AU - Tang , Tao
AU - Zhou , Tao
JO - Journal of Computational Mathematics
VL - 2
SP - 325
EP - 344
PY - 2023
DA - 2023/02
SN - 41
DO - http://doi.org/10.4208/jcm.2207-m2022-0020
UR - https://global-sci.org/intro/article_detail/jcm/21439.html
KW - Diffusion equations, Variable-step third-order BDF scheme, Discrete gradient structure, Discrete orthogonal convolution kernels, Stability and convergence.
AB - <p style="text-align: justify;">This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see, e.g., [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the $L^2$ norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.</p>