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Analysis of adaptive BDF2 scheme for diffusion equations. (English) Zbl 1466.65067
Summary: The variable two-step backward differentiation formula (BDF2) is revisited via a new theoretical framework using the positive semi-definiteness of BDF2 convolution kernels and a class of orthogonal convolution kernels. We prove that, if the adjacent time-step ratios \(r_k:=\tau_k/\tau_{k-1}\le (3+\sqrt{17})/2\approx 3.561\), the adaptive BDF2 time-stepping scheme for linear reaction-diffusion equations is unconditionally stable and (maybe, first-order) convergent in the \(L^2\) norm. The second-order temporal convergence can be recovered if almost all of time-step ratios \(r_k\le 1+\sqrt{2}\) or some high-order starting scheme is used. Specially, for linear dissipative diffusion problems, the stable BDF2 method preserves both the energy dissipation law (in the \(H^1\) seminorm) and the \(L^2\) norm monotonicity at the discrete levels. An example is included to support our analysis.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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[1] Becker, J., A second order backward difference method with variable steps for a parabolic problem, BIT, 38, 4, 644-662 (1998) · Zbl 0923.65050
[2] Chen, Wenbin; Wang, Xiaoming; Yan, Yue; Zhang, Zhuying, A second order BDF numerical scheme with variable steps for the Cahn-Hilliard equation, SIAM J. Numer. Anal., 57, 1, 495-525 (2019) · Zbl 1435.65142
[3] Crouzeix, M.; Lisbona, F. J., The convergence of variable-stepsize, variable-formula, multistep methods, SIAM J. Numer. Anal., 21, 3, 512-534 (1984) · Zbl 0542.65038
[4] Emmrich, Etienne, Stability and error of the variable two-step BDF for semilinear parabolic problems, J. Appl. Math. Comput., 19, 1-2, 33-55 (2005) · Zbl 1082.65086
[5] Gear, C. W.; Tu, K. W., The effect of variable mesh size on the stability of multistep methods, SIAM J. Numer. Anal., 11, 1025-1043 (1974) · Zbl 0292.65041
[6] Grigorieff, Rolf Dieter, Stability of multistep-methods on variable grids, Numer. Math., 42, 3, 359-377 (1983) · Zbl 0554.65051
[7] Hairer, E.; N\o rsett, S. P.; Wanner, G., Solving Ordinary Differential Equations. I, Springer Series in Computational Mathematics 8, xvi+528 pp. (1993), Springer-Verlag, Berlin · Zbl 0789.65048
[8] Hundsdorfer, Willem; Ruuth, Steven J.; Spiteri, Raymond J., Monotonicity-preserving linear multistep methods, SIAM J. Numer. Anal., 41, 2, 605-623 (2003) · Zbl 1050.65070
[9] Le Roux, Marie-No\"{e}lle, Variable step size multistep methods for parabolic problems, SIAM J. Numer. Anal., 19, 4, 725-741 (1982) · Zbl 0483.65033
[10] Liao, Hong-Lin; Sun, Zhi-Zhong, Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations, Numer. Methods Partial Differential Equations, 26, 1, 37-60 (2010) · Zbl 1196.65154
[11] Liao, Hong-lin; Lyu, Pin; Vong, Seakweng, Second-order BDF time approximation for Riesz space-fractional diffusion equations, Int. J. Comput. Math., 95, 1, 144-158 (2018) · Zbl 1387.65088
[12] LiaoJiZhang:2019 H.-L. Liao, B. Ji and L. Zhang, An adaptive BDF2 implicit time-stepping method for the phase field crystal model, IMA L. Numer. Anal., 2020. DOI 10.1093/imanum/draa075.
[13] Liao, Hong-lin; Tang, Tao; Zhou, Tao, On energy stable, maximum-principle preserving, second-order BDF scheme with variable steps for the Allen-Cahn equation, SIAM J. Numer. Anal., 58, 4, 2294-2314 (2020) · Zbl 1447.65083
[14] Nishikawa, Hiroaki, On large start-up error of BDF2, J. Comput. Phys., 392, 456-461 (2019) · Zbl 1452.65126
[15] Shampine, Lawrence F.; Reichelt, Mark W., The MATLAB ODE suite, SIAM J. Sci. Comput., 18, 1, 1-22 (1997) · Zbl 0868.65040
[16] Thom\'{e}e, Vidar, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics 25, xii+370 pp. (2006), Springer-Verlag, Berlin · Zbl 1105.65102
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