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
N1 - H. L. Liao is supported by NSF of China under grant number 12071216, T. Tang is supported by NNW2018-ZT4A06 project and T Zhou is supported by NSF of China under grant numbers 12288201 and youth innovation promotion association (CAS).
Publisher Copyright:
© 2023 Global Science Press. All rights reserved.
PY - 2023/3
Y1 - 2023/3
N2 - 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 L2norm 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.
AB - 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 L2norm 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.
KW - Diffusion equations
KW - Discrete gradient structure
KW - Discrete orthogonal convolution kernels
KW - Stability and convergence
KW - Variable-step third-order BDF scheme
UR - http://www.scopus.com/inward/record.url?scp=85152747273&partnerID=8YFLogxK
U2 - 10.4208/jcm.2207-m2022-0020
DO - 10.4208/jcm.2207-m2022-0020
M3 - Journal article
AN - SCOPUS:85152747273
SN - 0254-9409
VL - 41
SP - 325
EP - 344
JO - Journal of Computational Mathematics
JF - Journal of Computational Mathematics
IS - 2
ER -