Title

STABILITY OF GENERALIZED BDF2 AND AM2 METHODS

Alternative Title

Author
Name pinyin
YU Yinqian
School number
12132890
Degree

Discipline
070102 计算数学
Subject category of dissertation
07 理学
Supervisor

Mentor unit

Publication Years
2023-05-17
Submission date
2023-06-28
University

Place of Publication

Abstract

The main part of this thesis is to develop a unified approach to the uniform-in-time energy stability of a special class of linear two-step numerical methods when applied to some prototype linear dissipative problems. Also, some classical stability properties of these multi-step methods are analyzed, including A-stability(linear stability), G-stability, stability region, and decay rate. The aim is to construct better multi-step methods based on the analysis of these classical properties. Starting from recalling several classical linear multi-step methods, including the classical Adams-Bashforth(AM) methods and Backward differentiation formula(BDF) methods, we borrow the idea of Talor’s expansion based on different time point 𝑡 ∗ in order to construct the generalized two-step methods. After that, considering the efficiency and stability of numerical methods, several stability properties are recalled. It is natural to explore the stability of the generalized two-step methods, especially the generalized second-order Adams-Bashforth(AM2) method and the generalized second-order Backward differentiation formula(BDF) method. The former is derived by the expansion centered on the mid-point of the current and future time grid, while the latter is centered on the future time grid. In terms of the formulas, the generalized AM2 and the generalized BDF2 method are obtained by introducing a free parameter in their classical version. Therefore, we expect the generalized method can have better properties than the classical one, i.e., a lower local error and better stability. As a result, we find that the generalized AM2 method and the generalized BDF2 method are sometimes more stable than their classical version, i.e., with a larger stability region and larger decay rate. Naturally, we expect to make full advantage of these generalized two-step methods in some problems involving long-time behavior such as the coarsening process for the Cahn-Hilliard and several phase-field thin-film models. Lots of past research about the long-time stability issue of numerical schemes motivates us to explore whether the generalized AM2 method and the generalized BDF2 method can inherit the uniform-in-time energy bound in some linear dissipative problems. We eventually have some theoretical results about the uniform-in-time energy stability of the generalized AM2 and the generalized BDF2 methods in some prototype linear dissipative systems with a symmetric positive definite operator and a mild anti-symmetric operator. Additionally, some numerical results are given to show the advantages of these methods in accuracy and stability. On the other hand, we also try to explore the relationship between several stability properties, especially A-stability and uniform-in-time energy stability. It shows that in some cases, A-stability has an equivalent relationship with uniform-in-time energy stability in linear two-step convergent methods. And it can be directly used in some simple linear dissipative models to obtain the unconditionally uniform-in-time energy stability

Other Abstract

Keywords
Language
English
Training classes

Enrollment Year
2021
Year of Degree Awarded
2023-06
References List

[1] M. ANITESCU, F. PAHLEVANI, AND W. J. LAYTON, Implicit for local effects and explicit for nonlocal effects is unconditionally stable, Electron. Trans. Numer. Anal, 18 (2004), pp. 174–187.
[2] U. M. ASCHER, S. J. RUUTH, AND B. T. WETTON, Implicit-explicit methods for time-dependent partial differential equations, SIAM Journal on Numerical Analysis, 32 (1995), pp. 797–823.
[3] W. CHEN, M. GUNZBURGER, D. SUN, AND X. WANG, Efficient and long-time accurate second-order methods for the Stokes–Darcy system, SIAM Journal on Numerical Analysis, 51 (2013), pp. 2563–2584.
[4] W. CHEN, W. LI, C. WANG, S. WANG, AND X. WANG, Energy stable higher-order linear ETD multi-step methods for gradient flows: application to thin film epitaxy, Research in the Mathematical Sciences, 7 (2020), pp. 1–27.
[5] Y. CHEN AND J. SHEN, Efficient, adaptive energy stable schemes for the incompressible Cahn–Hilliard Navier–Stokes phase-field models, Journal of Computational Physics, 308 (2016), pp. 40–56.
[6] B. COCKBURN, D. JONES, AND E. TITI, Estimating the number of asymptotic degrees of freedomfor nonlinear dissipative systems, Mathematics of Computation, 66 (1997), pp. 1073–1087.
[7] C. W. CRYER, A new class of highly-stable methods: A_0-stable methods, BIT Numerical Mathematics, 13 (1973), pp. 153–159.
[8] G. DAHLQUIST, Convergence and stability in the numerical integration of ordinary differential equations, Mathematica Scandinavica, (1956), pp. 33–53.
[9] G. G. DAHLQUIST, A special stability problem for linear multistep methods, BIT Numerical Mathematics, 3 (1963), pp. 27–43.
[10] DAHLQUIST, GERMUND, Error analysis for a class of methods for stiff non-linear initial value problems, in Numerical Analysis, Springer, 1976, pp. 60–72.
[11] DAHLQUIST, GERMUND, G-stability is equivalent to A-stability, BIT Numerical Mathematics, 18 (1978), pp. 384–401.
[12] DAHLQUIST, GERMUND, 33 years of numerical instability, Part I, BIT Numerical Mathematics, 25 (1985), pp. 188–204.
[13] C. FOIAS, G. R. SELL, AND E. S. TITI, Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations, Journal of Dynamics and Differential Equations, 1 (1989), pp. 199–244.
[14] C. W. GEAR, Numerical initial value problems in ordinary differential equations, Prentice-Hall Series in Automatic Computation, (1971).
[15] S. GOTTLIEB, F. TONE, C. WANG, X. WANG, AND D. WIROSOETISNO, Long time stability of a classical efficient scheme for two-dimensional Navier–Stokes equations, SIAM Journal on Numerical Analysis, 50 (2012), pp. 126–150.E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving ordinary differential equations I, Nonstiff problems, Springer-Vlg, 1993.
[17] P. HENRICI, Discrete variable methods in ordinary differential equations, New York: Wiley,(1962).
[18] A. ISERLES, A first course in the numerical analysis of differential equations, no. 44, CambridgeUniversity Press, 2009.
[19] R. JELTSCH, d its relation to A_0-and A(0)-Stability, SIAM Journal on Numerical Analysis, 13(1976), pp. 8–17.
[20] R. JELTSCH AND O. NEVANLINNA, Stability and accuracy of time discretizations for initial value problems, Numerische Mathematik, 40 (1982), pp. 245–296.
[21] N. JIANG, M. MOHEBUJJAMAN, L. G. REBHOLZ, AND C. TRENCHEA, An optimally accurate discrete regularization for second order time stepping methods for Navier–Stokes equations, Computer Methods in Applied Mechanics and Engineering, 310 (2016), pp. 388–405.
[22] W. LAYTON AND C. TRENCHEA, Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations, Applied Numerical Mathematics, 62 (2012), pp. 112–120.
[23] W. E. MILNE, Numerical integration of ordinary differential equations, The American Mathematical Monthly, 33 (1926), pp. 455–460.
[24] W. E. MILNE AND W. MILNE, Numerical solution of differential equations, vol. 64, Wiley New York, 1953.
[25] A. R. MITCHELL AND J. W. CRAGGS, Stability of difference relations in the solution of ordinary differential equations, Mathematics of Computation, 7 (1953), pp. 127–129.
[26] O. NEVANLINNA AND W. LINIGER, Contractive methods for stiff differential equations part ii, BIT Numerical Mathematics, 19 (1979), pp. 53–72.
[27] E. J. NYSTRÖM, Über die numerische Integration von Differentialgleichungen, akademische Abhandlung... von EJ Nyström, Druckerei der finnischen Literaturgesellschaft, 1925.
[28] H. ROBERTSON, The solution of a set of reaction rate equations, Numerical Analysis: An Introduction, 178182 (1966).
[29] H. RUTISHAUSER, Über die instabilität von methoden zur integration gewöhnlicher differentialgleichungen, Zeitschrift für angewandte Mathematik und Physik ZAMP, 3 (1952), pp. 65–74.
[30] J. SHEN, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Applicable Analysis, 38 (1990), pp. 201–229.
[31] B. SOMMEIJER, Explicit, high-order Runge-Kutta-Nyström methods for parallel computers, Applied Numerical Mathematics, 13 (1993), pp. 221–240.
[32] A. STUART AND A. R. HUMPHRIES, Dynamical systems and numerical analysis, vol. 2, Cambridge University Press, 1998.
[33] A. M. STUART, Numerical analysis of dynamical systems, Acta Numerica, 3 (1994), pp. 467–572.
[34] J. TODD, Notes on modern numerical analysis. I. Solution of differential equations by recurrence relations, Mathematics of Computation, 4 (1950), pp. 39–44.
[35] G. WANNER AND E. HAIRER, Solving ordinary differential equations II, vol. 375, Springer Berlin Heidelberg New York, 1996.
[36] O. B. WIDLUND, A note on unconditionally stable linear multistep methods, BIT Numerical Mathematics, 7 (1967), pp. 65–70.

Academic Degree Assessment Sub committee

Domestic book classification number
O241
Data Source

Document TypeThesis
Identifierhttp://kc.sustech.edu.cn/handle/2SGJ60CL/544405
DepartmentDepartment of Mathematics
Recommended Citation
GB/T 7714
Yu YQ. STABILITY OF GENERALIZED BDF2 AND AM2 METHODS[D]. 深圳. 南方科技大学,2023.
 Files in This Item: File Name/Size DocType Version Access License 12132890-于寅乾-数学系.pdf（1402KB） Restricted Access -- Fulltext Requests
 Related Services Fulltext link Recommend this item Bookmark Usage statistics Export to Endnote Export to Excel Export to Csv Altmetrics Score Google Scholar Similar articles in Google Scholar [于寅乾]'s Articles Baidu Scholar Similar articles in Baidu Scholar [于寅乾]'s Articles Bing Scholar Similar articles in Bing Scholar [于寅乾]'s Articles Terms of Use No data! Social Bookmark/Share
All comments (0)
No comment.

Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.