In this paper we shall study stability and error estimates for time discretizations by the two-step backward differentiation formula (BDF2) with variable step-sizes for linear parabolic partial differential equations (PDEs)
and its semilinear extension, where : is a positive definite, self-adjoint, linear operator on a Hilbert space with domain dense in , the linear operator satisfies some structural assumptions; here the forcing term , and initial value .
Let the time interval for given , , be partitioned via . Let , be the time step-sizes which in general will be variable, and . We set
Assuming we are given starting approximations and , which is computed by the trapezoidal method or the backward Euler scheme, we discretize (1.1) in time by the variable step-size BDF2, i.e., we define nodal approximations to the values of the exact solution to (1.1) as follows:
Here and are defined by
respectively. For an equidistant partition with , we have and the well-known formula
The BDF2 method is one of the most popular time-stepping methods and many studies have been conducted on the stability and error estimates for it. Because of its good stability property (the scheme is -stable), the BDF2 method with constant step-size has been dealt with for various equations as, e.g., linear parabolic equations [24, 4], integro-differential equations , jump-diffusion model in finance , the Navier-Stokes equations [15, 20, 14]. When the solutions of time dependent differential equations have different time scales, i.e., solutions rapidly varying in some regions of time while slowly changing in other regions, variable step-sizes are often essential to obtain computationally efficient, accurate results. Owing to these prominent advantages, the variable step-size BDF2 method has been successfully applied to partial integro-differential equations  and Cahn-Hilliard equation  recently. An important result that the variable step-size BDF2 method is zero-stable if the step-size ratios are less than has been independently proved by several authors [26, 16, 10]. And the value of cannot be improved when dealing with arbitrary variable step-sizes (see, for example, [16, 6, 7]).
For the variable step-size BDF2 method applied to linear parabolic equations, Le Roux  derived stability and error bounds in the norm by using spectral techniques under the step-size conditions
with constants and . Palencia and García-Archilla  studied linear parabolic equations in a Banach space setting and obtained that the ratios should be close to such as, e.g., in (3) for the stability factor to be moderate. Grigorieff [17, 18] showed stability and optimal error estimates with smooth or non-smooth data under the assumption that the step-size ratios are less than in a Banach space setting. Becker improved the bound up to in Hilbert space  (see, also, ) by testing (2) with for a specified constant . Based on the same technique, i.e., testing (2) with , Emmrich  extended the results to semilinear parabolic problems and further improved the bound to using a more general identity for . Emmrich  also studied the stability and convergence of the variable step-size BDF2 method for nonlinear evolution problems governed by a monotone potential operator.
It is natural to ask what the upper bound of step-size ratios is and whether it is identical with the upper bound for the zero-stability. In this paper we will address this question and give an affirmative answer. We explore a new technique, which is very different from the one used by Becker , Thomée  and Emmrich [12, 13]. We first test (2) with to obtain -stability and -stability (their definitions will be introduced in Section 2). Then after we test (2) with , using -stability estimate, we obtain the usual stability result in the norm under the sharp zero-stability condition on the ratios of consecutive step-sizes. Following the approach of Chen et. al. , the and -stabilities of the variable step-size BDF2 method are also established under a more relaxed assumption that the step-size ratios are less than .
It is well known that the method (2) yields second order approximations to (in the norm) when the backward Euler method is used to compute the starting value , since it is applied only once. This choice for is quite popular in the multistep methods for computations of parabolic equations. However, the error bounds derived in this paper based on the obtained stability results suggest that this is not the best choice for the constant step-size BDF2 method, since it will cause the reduction of the convergence order in the and norms. This will be discussed in detail in Section 4.
The rest of this paper is organized as follows. We start in Section 2 by introducing the necessary notation and recalling a lemma which will be used in the following analysis. The stability of the method in several norms under the condition that the step-size ratios are less than (or ) is proved in Section 3. Error estimates in different norms are derived in Section 4. Since our error estimates will depend on the first step error , the error produced by the trapezoidal method or the backward Euler scheme will be analyzed in this section too. Section 5 will extend the analysis to the semilinear case
with some assumptions on the nonlinear operator . Section 6 is devoted to numerical experiments, which confirm our theoretical results and illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations. Section 7 contains a few concluding remarks.
2 Variable step-size BDF2 method for linear parabolic equations
Now we consider the variable two-step BDF method for solving (1.1). To do this, we first make some assumptions and introduce the necessary notation.
2.1 Linear parabolic equations
Let and denote the norms in and by and , , respectively. Let be the dual of (), and denote by the dual norm on , . We denote by the duality pairing between and . We define a bilinear form via . For the linear operator , we assume that
with a smooth nonnegative function . Let . We may write the parabolic problem in variational form as
Standard example. Let and be defined by
respectively, where are sufficiently smooth functions in with being a bounded domain in with sufficiently smooth boundary . Let and be the usual Sobolev and Lebesgue space, respectively. Assume that is symmetric and uniformly positive definite. Then the operators is a positive definite, self-adjoint, linear operator, and satisfies the condition (5).
2.2 Variable step-size BDF2 method for linear parabolic equations
For the method (2) we need the starting values and . We set and perform an initial trapezoidal approximation to get
with . Note that with , the two-step BDF formally degenerates to a backward Euler step. It is also easy to see that
With respect to the solvability of the time discrete problem, Emmrich has shown in  that for given and , the problem
admits a unique solution. For the obtained solution sequence , we define the , and norms as
respectively. It is well known that they are the discrete counterparts of the , and norms, respectively.
Remark. [The choice for ]. The starting value can be also obtained by the backward Euler method
with . It is well known that the constant step-size BDF2 method (2) with this initial approximation also yields second order approximations to in norm; see Corollary 4.6 in Section 4, or, [5, 24]. However, we find that order reduction will be caused in the and norms when the starting value is obtained by the backward Euler method (12). Here the norm of a solution sequence with is defined by
which is the discrete counterpart of with nonuniform grid weights .
Because of the different choices for and , we pay special attention to and set .
In subsequent sections, by convention, we set and if . We will use the identity
We also need the following discrete Gronwall lemma proved in .
[Discrete Gronwall lemma ] Let , and with being monotonically increasing. Then
3 Stability analysis
In this section we shall show stability of the variable step-size BDF2 method with applied to linear parabolic equations (1.1). As mentioned in Introduction, for the variable step-size BDF2 method applied to parabolic equations, the best known result is that it is stable in the norm when the step-size ratios are less than . To improve the bound to , the upper bound for the zero-stability, we first need the following stability results in the norm. Additionally, since the norm is an energy norm, from a physical point of view, stability is of utmost important.
[ and stability under ] Let with . If there exists a constant such that satisfies
then we have, for any ,
Summing up gives for ,
Now take such that
Then we get
An application of Lemma 2.2 leads to the desired result.
Recently, the stability of a linearly implicit stabilization BDF2 method with variable step-sizes has been established under the condition for the Cahn-Hilliard equation in . Following their approach, we can also improve the bound to for the stability of the variable step-size BDF2 method for the problem (1.1).
[ and stability under ] Let with , and let . If there exists a constant such that satisfies
then we have, for any ,
Taking in (18) the inner product with , we obtain, for any ,
Ignoring some of the positive terms on the left-hand side, we have
where . In the case , noting that , we can take such that
In the case , since is a decreasing function, we take such that (27) holds. Thus in both cases, we have
Now we use the stability estimate (17) in Theorem 3.1 to show the stability of the variable step-size BDF2 method in the and norms.
[ and stability under ] Let with . If there exist constants and such that satisfies (16) and
where , then the following estimate holds for :
Here, depends on , , , and , , with being defined by
Taking in (2) the inner product with yields
By simple calculations, the first term on the left-hand side becomes