In the past decade, many numerical methods have been proposed to solve the time-fractional diffusion equation. Some of these methods are on the uniform time meshes. For example, the L1 scheme of -order has been well-developed by Langlands and Henry , Sun-Wu , and Lin-Xu , etc. Alikhanov  proposed the L2-1 scheme that has second order accuracy in time. An L2 method of -order on uniform meshes is studied in  by Gao-Sun-Zhang. In , a slightly different L2 fractional-derivative operator is analyzed by Lv-Xu for uniform meshes, where the optimal convergence -order in time is obtained under strong regularity assumptions on the exact solution.
Recently, those methods on nonuniform time meshes for time-fractional diffusion equation have attracted more and more attention, in particular, on the graded meshes. In fact, the exact solution to the time-fractional diffusion equation could have low regularity in general near the initial time , which would deteriorate the convergence rate of the numerical solutions. This motivates researchers to consider nonuniform time meshes to obtain the desired -order convergence rate under low regularity assumptions on the exact solution. For example, Stynes-Riordan-Gracia  prove the sharp error analysis of L1 scheme on graded meshes. Kopteva provides a different framework for the analysis of the L1 scheme on graded meshes in two and three spatial dimensions in , and analyzes an L2 scheme of -order on graded meshes in . Note that in these works, the -stability of numerical solutions is established.
In addition to the L1 and L2 methods on graded meshes, we mention the convolution quadrature methods with corrections which can also overcome the convergence rate problem for time-fractional diffusion equation, see for example [4, 5] and the references therein.
In this work, we consider the -stability of an L2 method (the same as ) on general nonuniform meshes for subdiffusion equation. Note that in general the time steps could change freely, unlike the graded meshes controlled by the grading parameter. For the L2 fractional-derivative operator denoted by , we prove that the following bilinear form
is positive semidefinite under mild restrictions on the time step ratio with ( is the th time step), see Theorem 3.2 for details. In particular, if
such positive semidefiniteness holds. As a consequence, the -stability of the L2 scheme for the subdiffusion equation with homogeneous Dirichlet boundary condition can be derived for all time, see Theorem 4.1. Moreover, in the special case of graded meshes, we show that if the grading parameter
then is positive semidefinite and consequently the -stability of L2 scheme is established.
This work is organized as follows. In Section 2, the derivation, explicit expression and reformulation of L2 fractional-derivative operator are provided. In Section 3, we prove the positive semidefiniteness of the bilinear form under some mild restrictions on the time step ratios. In Section 4, we establish the -stability of the L2 scheme for the subdiffusion equation, based on the positive semidefiniteness result. In Section 5, the special case of graded meshes is discussed.
2. Discrete fractional-derivative operator
In this part we show the derivation, explicit expression and reformulation of L2 operator on an arbitrary nonuniform mesh.
We consider the L2 approximation of the fractional-derivative operator defined by
Take a nonuniform time mesh with . When
, we use the standard linear Lagrangian polynomial interpolating:
When , for , we use the standard quadratic Lagrangian polynomial interpolating :
while for , we use the quadratic Lagrangian polynomial defined in (2.3).
Let . At , the fractional derivative is approximated by the discrete fractional-derivative operator
where for ,
It can be verified that , , for , and , , . Furthermore,
always holds for .
Specifically speaking, we can figure out the explicit expressions of and as follows (note that ): for ,
while for ,
We reformulate the discrete fractional derivative in (2.4) as
To establish the -stability of L2-type method for fractional-order parabolic problem, we want to prove the positive semidefiniteness of the following bilinear form
where is the standard inner product over and are two functions defined on and . That is to say, we want to prove that for any defined on and ,
3. Positive semidefiniteness of bilinear form
3.1. Properties of L2 coefficients
We propose the following properties of the L2 coefficients , and in (2.12), which will be useful to establish the positive semidefiniteness of bilinear form .
Lemma 3.1 (Properties of , and ).
We first provide two equivalent forms of in (2.5) as follows:
It is easy to see , i.e., (P1) holds.
i.e., the properties (P2) and (P4) hold.
We now turn to prove the properties of and . For in (2.5), we have
This is the property (P5). Since , we have for and the property (P7) holds. For any fixed ,
decreases w.r.t. , implying that
i.e., the property (P6). Combining this with property (P2), the property (P8) holds.
Note that for any fixed ,
decreases w.r.t. , and
Using (3.14) and the fact
we can derive from (3.10) that
To ensure the property (P9), we need
that is the condition (3.1).