Backward Difference Formulae: The Energy Technique for Subdiffusion Equation

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.

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

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