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Backward Difference Formulae: The Energy Technique for Subdiffusion Equation

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Abstract

Based on the equivalence of A-stability and G-stability, the energy technique of the six-step BDF method for heat equation has been discussed in [Akrivis, Chen, Yu, Zhou, SIAM J. Numer. Anal., Minor Revised]. Unfortunately, this theory is hard to apply in the time-fractional PDEs. In this work, we consider three types of subdiffusion models, namely single-term, multi-term and distributed order fractional diffusion equations. We present a novel and concise stability analysis of the time stepping schemes generated by k-step backward difference formulae (BDFk), for approximately solving the subdiffusion equation. The analysis mainly relies on the energy technique by applying Grenander-Szegö theorem. This kind of argument has been widely used to confirm the stability of various A-stable schemes (e.g., \(k=1,2\)). However, it is not an easy task for higher order BDF methods, due to lack of the A-stability. The core object of this paper is to fill in this gap.

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References

  1. Akrivis, G.: Stability of implicit and implicit-explicit multistep methods for nonlinear parabolic equations. IMA J. Numer. Anal. 38, 1768–1796 (2018)

    Article  MathSciNet  Google Scholar 

  2. Akrivis, G., Chen, M.H., Yu, F., Zhou, Z.: The energy technique for the six-step BDF method. SIAM J. Numer. Anal. (Minor Revised) arXiv:2007.08924

  3. Akrivis, G., Katsoprinakis, E.: Backward difference formulae: new multipliers and stability properties for parabolic equations. Math. Comp. 85, 2195–2216 (2016)

  4. Baiocchi, C., Crouzeix, M.: On the equivalence of A-stability and G-stability. Appl. Numer. Math. 5, 19–22 (1989)

    Article  MathSciNet  Google Scholar 

  5. Carmi, S., Barkai, E.: Fractional Feynman–Kac equation for weak ergodicity breaking. Phys. Rev. E 84, 061104 (2011)

    Article  Google Scholar 

  6. Carmi, S., Turgeman, L., Barkai, E.: On distributions of functionals of anomalous diffusion paths. J. Stat. Phys. 141, 1071–1092 (2010)

    Article  MathSciNet  Google Scholar 

  7. Chan, R.H.F., Jin, X.Q.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007)

    Book  Google Scholar 

  8. Chen, M.H., Deng, W.H.: Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators. Commun. Comput. Phys. 16, 516–540 (2014)

    Article  MathSciNet  Google Scholar 

  9. Chen, M.H., Deng, W.H.: Fourth order accurate scheme for the space fractional diffusion equations. SIAM J. Numer. Anal. 52, 1418–1438 (2014)

    Article  MathSciNet  Google Scholar 

  10. Chen, M.H., Deng, W.H.: Discretized fractional substantial calculus. ESAIM M2AN 49, 373–394 (2015)

  11. Chen, M.H., Deng, W.H.: High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights. SIAM J. Sci. Comput. 37, A890–A917 (2015)

    Article  Google Scholar 

  12. Chen, M.H., Deng, W.H.: High order algorithm for the time-tempered fractional Feynman–Kac equation. J. Sci. Comput. 76, 867–887 (2018)

    Article  MathSciNet  Google Scholar 

  13. Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 75, 673–696 (2006)

  14. Dahlquist, G.: G-stability is equivalent to A-stability. BIT 18, 384–401 (1978)

    Article  MathSciNet  Google Scholar 

  15. Gao, G.H., Sun, H.H., Sun, Z.Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equation based on certain superconvergence. J. Comput. Phys. 280, 510–528 (2015)

    Article  MathSciNet  Google Scholar 

  16. Friedrich, R., Jenko, F., Baule, A., Eule, S.: Anomalous diffusion of inertial, weakly damped particles. Phys. Rev. Lett. 96, 230601 (2006)

    Article  Google Scholar 

  17. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (2010)

    MATH  Google Scholar 

  18. Henrici, P.: Applied and Computational Complex Analysis. New York (1974)

  19. Ji, C.C., Sun, Z.Z.: A high-order compact finite difference schemes for the fractional sub-diffusion equation. J. Sci. Comput. 64, 959–985 (2015)

    Article  MathSciNet  Google Scholar 

  20. Jin, B.T., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, A146–A170 (2016)

    Article  MathSciNet  Google Scholar 

  21. Jin, B.T., Lazarov, Thomée, T, R., Zhou, Z.: On nonnegativity preservation in finite element methods for subdiffusion equations. Math. Comp. 86, 2239–2260 (2017)

  22. Jin, B.T., Li, B.Y., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39, A3129–A3152 (2017)

    Article  MathSciNet  Google Scholar 

  23. Klafter, J., Sokolov, I.M.: First Steps in Random Walks: From Tools to Applications. Oxford University Press, New York (2011)

    Book  Google Scholar 

  24. Li, B.Y., Wang, K., Zhou, Z.: Long-time accurate symmetrized implicit-explicit BDF methods for a class of parabolic equations with non-self-adjoint operators. SIAM J. Numer. Anal. 58, 189–210 (2020)

    Article  MathSciNet  Google Scholar 

  25. Li, C.P., Ding, H.F.: Higher order finite difference method for the reaction and anomalous-diffusion equation. Appl. Math. Modell. 38, 3802–3821 (2014)

    Article  MathSciNet  Google Scholar 

  26. Liu, J.: Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the stokes equations on time varying domains. SIAM J. Numer. Anal. 51, 743–772 (2013)

    Article  MathSciNet  Google Scholar 

  27. Lubich, C., Mansour, D., Venkataraman, C.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 33, 1365–1385 (2013)

    Article  MathSciNet  Google Scholar 

  28. Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)

    Article  MathSciNet  Google Scholar 

  29. Lubich, C.: Convolution quadrature and discretized operational calculus I. Numer. Math. 52, 129–145 (1988)

    Article  MathSciNet  Google Scholar 

  30. Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comp. 65, 1–17 (1996)

    Article  MathSciNet  Google Scholar 

  31. Nevanlinna, O., Odeh, F.: Multiplier techniques for linear multistep methods. Numer. Funct. Anal. Optim. 3, 377–423 (1981)

    Article  MathSciNet  Google Scholar 

  32. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  33. Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, Berlin (2007)

    Book  Google Scholar 

  34. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)

    Article  MathSciNet  Google Scholar 

  35. Shi, J.K., Chen, M.H.: Correction of high-order BDF convolution quadrature for fractional Feynman-Kac equation with Lévy flight. J. Sci. Comput. 85, 28 (2020)

    Article  Google Scholar 

  36. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  37. Xu, D.: Uniform \(l^1\) behaviour in a second-order difference-type method for a linear Volterra equation with completely monotonic kernel I:stability. IMA J. Numer. Anal. 31, 1154–1180 (2011)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Minghua Chen & Fan Yu

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China

    Zhi Zhou

Authors
  1. Minghua Chen
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  2. Fan Yu
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  3. Zhi Zhou
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Correspondence to Minghua Chen.

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This work was supported by NSFC 11601206 and Hong Kong RGC grant (No. 25300818).

Appendix

Appendix

Let

$$\begin{aligned} g(\zeta )=\left( \sum _{j=1}^{k}\frac{1}{j}(1-e^{-\sigma \tau }\zeta )^{j}\right) ^\alpha =\sum _{j=0}^{\infty }g_j^{(k)}{\zeta }^{j},\quad g_j^{(k)}=e^{-\sigma j \tau }l_j^{(k)}. \end{aligned}$$

Then the coefficients \(\{l_j^{(k)}\}_{j=0}^{\infty }\) are given explicitly by the following recurrence relation, see Theorem 1.6 of [18].

  • BDF1

    $$\begin{aligned} l_0^{(k)}=1,~l_j^{(k)}=\left( 1-\frac{{\alpha }+1}{j}\right) l_{j-1}^{(k)},~j\geqslant 1. \end{aligned}$$

  • BDF2

    $$\begin{aligned} \begin{aligned} l_0^{(k)}&=\left( \frac{3}{2}\right) ^{\alpha },~l_1^{(k)}=-\left( \frac{3}{2}\right) ^{\alpha }\frac{4}{3}{\alpha },\\ l_j^{(k)}&=\frac{4}{3}\left( 1-\frac{{\alpha }+1}{j}\right) l_{j-1}^{(k)}+\frac{1}{3}\left( \frac{2({\alpha }+1)}{j}-1\right) l_{j-2}^{(k)},~j\geqslant 2. \end{aligned} \end{aligned}$$

  • BDF3

    $$\begin{aligned} \begin{aligned} l_0^{(k)}&=\left( \frac{11}{6}\right) ^{\alpha },~l_1^{(k)}=-\left( \frac{11}{6}\right) ^{\alpha }\frac{18}{11}{\alpha },~ l_2^{(k)}=\left( \frac{11}{6}\right) ^{\alpha }\left( \frac{162}{121}{\alpha }^2-\frac{63}{121}{\alpha }\right) ,\\ l_j^{(k)}&=\frac{18}{11}\left( 1-\frac{{\alpha }+1}{j}\right) l_{j-1}^{(k)}\!+\!\frac{18}{22}\left( \frac{2({\alpha }+1)}{j}-1\right) \!l_{j-2}^{(k)}\\&\quad +\frac{2}{11}\left( 1-\frac{3\left( {\alpha }+1\right) }{j}\right) \!l_{j-3}^{(k)},~j\geqslant 3. \end{aligned} \end{aligned}$$

  • BDF4

    $$\begin{aligned} \begin{aligned} l_0^{(k)}&=\left( \frac{25}{12}\right) ^{\alpha },~l_1^{(k)}=-\left( \frac{25}{12}\right) ^{\alpha }\frac{48}{25}{\alpha },~ l_2^{(k)}=\left( \frac{25}{12}\right) ^{\alpha }\left( \frac{1152}{625}{\alpha }^2-\frac{252}{625}{\alpha }\right) ,\\ l_3^{(k)}&=\left( \frac{25}{12}\right) ^{\alpha }\left( -\frac{18432}{15625}{\alpha }^3+\frac{12096}{15625}{\alpha }^2-\frac{3664}{15625}{\alpha }\right) ,\\ l_j^{(k)}&=\frac{48}{25}\left( 1-\frac{{\alpha }+1}{j}\right) l_{j-1}^{(k)}+\frac{36}{25}\left( \frac{2({\alpha }+1)}{j}-1\right) l_{j-2}^{(k)}\\&\quad +\frac{16}{25}\left( 1-\frac{3\left( {\alpha }+1\right) }{j}\right) l_{j-3}^{(k)}+\frac{3}{25}\left( \frac{4({\alpha }+1)}{j}-1\right) l_{j-4}^{(k)},~j\geqslant 4. \end{aligned} \end{aligned}$$

  • BDF5

    $$\begin{aligned} \begin{aligned} l_0^{(k)}&=\!\left( \!\frac{137}{60}\!\right) ^{\alpha },~l_1^{(k)}=\!-\!\left( \!\frac{137}{60}\!\right) ^{\alpha }\frac{300}{137}{\alpha }\!,~ l_2^{(k)}\!=\!\left( \!\frac{137}{60}\!\right) ^{\alpha }\!\left( \!\frac{45000}{18769}{\alpha }^2\!-\!\frac{3900}{18769}{\alpha }\!\right) \!,\\ l_3^{(k)}&=\left( \frac{137}{60}\right) ^{\alpha }\left( -\frac{4500000}{2571353}{\alpha }^3+\frac{1170000}{2571353}{\alpha }^2-\frac{423800}{2571353}{\alpha }\right) ,\\ l_4^{(k)}&=\!\left( \!\!\frac{137}{60}\!\right) ^{\alpha }\!\left( \frac{337500000}{352275361}{\alpha }^4\!\!-\!\!\frac{175500000}{352275361}{\alpha }^3\! \!+\!\!\frac{134745000}{352275361}{\alpha }^2\!\!-\!\!\frac{\!103893525\!}{\!352275361\!}{\alpha }\!\!\right) ,\\ l_j^{(k)}&=\!\frac{300}{137}\!\left( \!1\!-\!\frac{\alpha +1}{j}\!\right) \!l_{j-1}^{(k)}\!+\!\frac{300}{137}\!\left( \!\frac{2({\alpha }+1)}{j}\!-1\!\right) \!l_{j-2}^{(k)} \!+\!\frac{200}{137}\!\left( \!1\!-\!\frac{3\!\left( \!{\alpha }\!+\!1\!\right) \!}{j}\!\right) \!l_{j-3}^{(k)}\\&\quad +\frac{75}{137}\left( \frac{4({\alpha }+1)}{j}-1\right) l_{j-4}^{(k)}+\frac{12}{137}\left( 1-\frac{5\left( {\alpha }+1\right) }{j}\right) l_{j-5}^{(k)},~j\geqslant 5. \end{aligned} \end{aligned}$$

  • BDF6

    $$\begin{aligned} \begin{aligned} l_0^{(k)}&=\!\left( \!\frac{147}{60}\!\right) ^{\alpha },~l_1^{(k)}\!=\!-\left( \frac{147}{60}\right) ^{\alpha }\frac{360}{147}{\alpha },~ l_2^{(k)}\!=\!\left( \!\frac{147}{60}\!\right) \!^{\alpha }\!\left( \!\frac{7200}{2401}{\alpha }^2\!+\!\frac{150}{2401}{\alpha }\!\!\right) ,\\ l_3^{(k)}&=\left( \frac{147}{60}\right) ^{\alpha }\left( -\frac{288000}{117649}{\alpha }^3-\frac{18000}{117649}{\alpha }^2-\frac{42400}{352947}{\alpha }\right) ,\\ l_4^{(k)}&=\left( \frac{147}{60}\right) ^{\alpha }\left( \frac{8640000}{5764801}{\alpha }^4+\frac{1080000}{5764801}{\alpha }^3 +\frac{1707250}{5764801}{\alpha }^2-\frac{2603575}{5764801}{\alpha }\right) ,\\ l_5^{(k)}&=\left( \frac{147}{60}\right) ^{\alpha }\left( -\frac{207360000}{282475249}{\alpha }^5-\frac{43200000}{282475249}{\alpha }^4-\frac{14730000}{40353607}{\alpha }^3\right. \\&\quad \left. +\frac{310309000}{282475249}{\alpha }^2-\frac{94994224}{282475249}{\alpha }\right) ,\\ l_j^{(k)}&=\!\frac{360}{147}\!\left( \!1\!-\!\frac{\alpha +1}{j}\!\right) \!l_{j-1}^{(k)}\!+\!\frac{450}{147}\!\left( \!\frac{2({\alpha }+1)}{j}\!-\!1\!\right) \!l_{j-2}^{(k)} \!+\!\frac{400}{147}\!\left( \!1\!-\!\frac{3\!\left( {\alpha }+1\right) \!}{j}\!\right) \!l_{j-3}^{(k)}\\&\quad +\frac{225}{147}\left( \frac{4({\alpha }+1)}{j}-1\right) l_{j-4}^{(k)}+\frac{72}{147}\left( 1-\frac{5\left( {\alpha }+1\right) }{j}\right) l_{j-5}^{(k)}\\&\quad +\frac{10}{147}\left( \frac{6({\alpha }+1)}{j}-1\right) l_{j-6}^{(k)},~j\geqslant 6. \end{aligned} \end{aligned}$$

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Chen, M., Yu, F. & Zhou, Z. Backward Difference Formulae: The Energy Technique for Subdiffusion Equation. J Sci Comput 87, 94 (2021). https://doi.org/10.1007/s10915-021-01509-9

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