Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations
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| Format: | Preprint |
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2022
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| author | Liao, Hong-lin Tang, Tao Zhou, Tao |
| author_facet | Liao, Hong-lin Tang, Tao Zhou, Tao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_12742 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations Liao, Hong-lin Tang, Tao Zhou, Tao Numerical Analysis 65M06, 65M12 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. |
| title | Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations |
| topic | Numerical Analysis 65M06, 65M12 |
| url | https://arxiv.org/abs/2204.12742 |