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Volume 41, Issue 2
Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations

Hong-lin Liao, Tao Tang & Tao Zhou

DOI: 10.4208/jcm.2207-m2022-0020

J. Comp. Math., 41 (2023), pp. 325-344.

Published online: 2023-02

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  • Abstract

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.

  • Keywords

Diffusion equations, Variable-step third-order BDF scheme, Discrete gradient structure, Discrete orthogonal convolution kernels, Stability and convergence.

  • AMS Subject Headings

65M06, 65M12.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

liaohl@nuaa.edu.cn (Hong-lin Liao)

tangt@sustech.edu.cn (Tao Tang)

tzhou@lsec.cc.ac.cn (Tao Zhou)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-325, author = {Liao , Hong-linTang , Tao and Zhou , Tao}, title = {Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {2}, pages = {325--344}, abstract = {

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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2207-m2022-0020}, url = {http://global-sci.org/intro/article_detail/jcm/21439.html} }
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 -

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.

Liao , Hong-linTang , Tao and Zhou , Tao. (2023). Discrete Energy Analysis of the Third-Order Variable-Step BDF Time-Stepping for Diffusion Equations. Journal of Computational Mathematics. 41 (2). 325-344. doi:10.4208/jcm.2207-m2022-0020
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