Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data

Two higher order time stepping methods based on the weighted and shifted Grünwald-Letnikov formulae in Tian et al. [34] are introduced and analyzed for the following subdiffusion problem, with \(0< \alpha <1\),where \(A=-\varDelta \) and \(\varDelta \) denotes the Laplacian defined on a regular domain \(\varOmega \subset {\mathbb {R}}^{d},d=1,2,3\) with smooth boundary \(\partial \varOmega \) and \(D(A)=H_{0}^{1}(\varOmega ) \cap H^{2}(\varOmega )\). Here the initial value \(u_{0} \in L_{2}(\varOmega )\) and the smoothness of the source term f is described in Theorems 1, 2 in Sect. 2 and 3, respectively. The time fractional derivative \(\, _{0}^{C} D_{t}^{\alpha } u(t), \, 0< \alpha <1 \) is defined in the sense of Caputo, see, e.g., Diethelm [10],More generally the operator A in (1) could be any linear, selfadjoint, positive definite operator with compact inverse, defined in \(D(A) \subset L_{2}(\varOmega )\), and satisfies the following resolvent estimates, with \(\pi /2< \theta _{0} < \pi \), see, e.g., Lubich et al. [23] and Thomée [33],It is easy to see that for any \( z \in \varSigma _{\theta } \) with \( \theta \in (\pi /2, \theta _{0})\), we have \(z^{\alpha } \in \varSigma _{\theta _{0}}\) since, with \(0< \alpha <1\),which implies that, by (2), see, e.g., Jin et al. [14, (2.3)],In Sects. 2 and 3, with some suitable approximation \(z_{k}\) of z, we shall choose \(\theta \in (\pi /2, \theta _{0}) \) sufficiently close to \(\pi /2\) such that \(z_{k}^{\alpha } \in \varSigma _{\theta _{0}}\) which guarantees that \((z_{k}^{\alpha } I + A)^{-1}\) exists.

Recently, Meerschaert et al. [27] and Tian et al. [34] introduced the weighted and shifted Gr\(\ddot{u}\)nwald-Letnikov difference operators to approximate the Riemann-Liouville fractional derivative and applied such difference operators to solve space fractional partial differential equations under the assumptions that the solution is sufficiently smooth and satisfies the homogeneous boundary conditions. To our knowledge, we have not seen any works in literature to apply such weighted and shifted Grünwald-Letnikov difference operator to construct higher order time discretization schemes for solving the subdiffusion equation (1). One of the possible reasons for lacking such works may be that the solution of (1) is not sufficiently smooth and it has the singularity near \(t=0\). For example, in the homogeneous case of (1) with \(f=0\), one has the following stability estimate, [29], with \(\Vert \cdot \Vert \) the norm in \(L_{2}(\varOmega )\),which shows that the \(\alpha \)-th order Caputo derivative of the solution of (1) becomes unbounded as \(t\rightarrow 0\). Hence, the \(C^{2}\)-regularity assumption, generally, does not hold for the exact solution of (1). Numerical experiments indicate that the convergence orders of some numerical methods for solving (1) actually do not hold uniformly in t even for the smooth data \(u_{0},\) see, e.g., Jin et al. [15], Stynes et al. [31] and Stynes [30]. Therefore, an attempt has been made in this paper to consider the higher order time discretization schemes for (1), based on the weighted and shifted Grünwald-Letnikov schemes developed in Tian et al. [34] and prove the optimal convergence orders of the proposed schemes with both smooth and nonsmooth data.

There are several approaches to improve the convergence orders of the numerical methods for solving (1), when the solution is not sufficiently smooth. One approach is to correct some weights of the numerical methods in order to capture the singularity of the solution. This idea was first introduced by Lubich et al. [23] for second order time stepping scheme applied to an evolution equation with positive memory term. After correcting some weights of the numerical methods, Lubich et al. [23] proved optimal convergence of the corrected numerical method for both smooth and nonsmooth data. Jin et al. [16] derived some higher order numerical methods in time for the problem (1), where the fractional derivatives are approximated by using the convolution quadrature generated by using the backward difference formulae. By correcting some starting steps of the numerical methods, Jin et al. [16] established that the corrected numerical methods have optimal order of convergence for any fixed time t for both smooth and nonsmooth data, see also [14, 17]. Subsequently, Yan et al. [36] corrected the starting steps of the L1 scheme for solving (1) and proved that the modified L1 scheme has the optimal convergence order \(O (k^{2-\alpha }), 0< \alpha <1\). More recently, Xing and Yan [35] analyzed a numerical method for solving (1), where the Caputo fractional derivative is expressed by using the Hadamard finite-part integral which is then approximated by using the quadratic interpolation polynomials. After correcting some starting steps and some weights of the high-order numerical methods, Xing and Yan [35] derived the optimal convergence order \(O(k^{3- \alpha }), 0< \alpha <1\) of the corrected numerical methods for both smooth and nonsmooth data. For the recent development of the corrections of numerical methods for (1), we refer the readers to the survey paper [12], see also [37]. For other numerical methods for solving time fractional diffusion equation, we refer to [3, 5‐8, 11, 15, 18‐22, 24, 25, 28, 32, 38‐40], etc.

The aim of this paper is to prove that the proposed numerical methods have the optimal convergence orders \(O(k^{2})\) and \(O(k^{3})\), respectively, by correcting a few starting steps of the numerical methods for both smooth and nonsmooth datas. Compared to other higher order time stepping methods in the literature for solving time diffusion problem (1), the proposed methods have two advantages: (i) The weights of our numerical methods are much simpler than those obtained by approximating the fractional derivative with the quadratic interpolating polynomials, see, e.g., in Xing and Yan [35] and further, these weights have a special structure as mentioned in Tian et al. [34], which may be useful for constructing some fast algorithms and also for proving their stability and error analyses; (ii) The weights of the proposed numerical schemes are related not only to the order of the fractional derivative, but also to the shifted numbers, which imply that our methods are more related to the equation itself, see, e.g., Tian et al. [34].

The main contributions of this paper are as follows. The paper is organized as follows. In Sect. 2, we consider the error estimates of the time discretization scheme for (1) with the convergence order \(O(k^{2})\) for both smooth and nonsmooth data. In Sect. 3, we derive the error estimates of the time discretization scheme with the convergence order \(O(k^{3})\) again for both smooth and nonsmooth data. Finally in Sect. 4, numerical examples are presented to show that numerical results are consistent with the theoretical results.

By C,  we denote a positive constant independent of discretization parameter k, but not necessarily the same at different occurrences.

In this section, we analyze a second order time discretization scheme for approximating the solution of the problem (1). After correcting some starting steps of the scheme, the optimal order of convergence is derived for the problem with both smooth and nonsmooth data.

Based on Tian et al. [34], we shall introduce a scheme to approximate the Riemann-Liouville fractional derivative \(_{0} ^{R}D^{\alpha }_{t}\phi (t)\). Let \(0=t_{0}<t_{1}<\cdots <t_{N}=T\) be a time partition of [0, T] and k be the step size. We define the following numerical scheme to approximate \(_{0} ^{R}D^{\alpha }_{t} \phi (t)\) at \(t=t_{n},n\ge 1\)whereHere \(g_{j}^{(\alpha )},j=0,1,2,\ldots ,\) are generated by the generating function \(\delta _{1}(\zeta )=(1-\zeta )\), that is,When \( p=0, \, q=-1\) or \(q=0, \, p=-1\), the equation (5) leads towhereIn Table 1, we show the differences numerically between the weights generated by (9) and by BDF2, i.e., backward difference formula with convergence order 2, in Jin et al. [16] with \(n=5\). Here \(w_{j}^{(\alpha )}\) and \(b_{j}^{(\alpha )}, 0< \alpha <1, \, j=0, 1, 2, \ldots , n\) denote the weights generated by (9) and by BDF2, respectively with respect to the different \(0<\alpha <1\).

The corresponding generating function \(\delta (\zeta )\) of the weights \(w^{(\alpha )}_{j}, j=0, 1, 2, \ldots \) in (9) satisfiesAs in Lubich et al. [23], we denote the discrete Laplace transform of the sequence \(w_{j}^{(\alpha )}, j=0, 1, 2, \ldots \) by \({\tilde{w}} (\zeta ) = \sum _{j=0}^{\infty } w_{j}^{(\alpha )} \zeta ^{j}\). We then have, by (10),The following lemma gives the series expansion of \({\tilde{w}} (\zeta )\) in (11) in terms of \((1- \zeta )\), see, the similar argument used in [16, (16)].

We now introduce a fully discrete scheme for solving (1). Let \({\mathcal {T}}_{h}\) denote a triangulation of \(\varOmega \) with h the maximal length of the sides on \({\mathcal {T}}_{h}\). Let \(S_{h}\subset H^{1}_0(\varOmega )\) denote the piecewise continuous linear finite element space.

For any fixed \(t \in (0, T]\), the finite element method of (1) is to find \(u_{h} (t) \in S_{h}\) such thatwhere \(A_{h}:S_{h}\rightarrow S_{h}\) denotes the discrete analogue of A defined with some suitable bilinear form \(a(\cdot ,\cdot )\) defined on \(H^1_0(\varOmega ) \times H^1_0(\varOmega )\) associated with the operator A, byand \(f_{h}=P_{h} f\), where \(P_{h}:L_{2}(\varOmega )\rightarrow S_{h}\) denotes the \(L_{2}\) project operator given byHere \(u_{0h} \in S_{h} \) denotes some approximation of \(u_{0}\in L_{2}(\varOmega )\). When \(u_{0}\) is nonsmooth, we choose \(u_{0h}=P_{h}u_{0}\) and when \(u_{0}\) is smooth, that is \(u_{0} \in D(A)\), we may choose \(u_{0h}=\mathbf{{R}}_{h}u_{0}\), where \(\mathbf{{R}}_{h}:H_{0}^{1}(\varOmega )\rightarrow S_{h}\) denotes the Ritz projection or elliptic projection defined byLet \(V_{h}(t)=u_{h}(t)-u_{0h}.\) Then, the equation (14) is equivalent toSince \(\, _{0} ^{C}D^{\alpha }_{t} V_{h}(t)=\, _{0} ^{R}D^{\alpha }_{t}( V_{h}(t)-V_{h}(0)),0<\alpha <1\), now (15) is rewritten asAn application of Taylor’s expansion as in Jin et al. [17] yieldswhere \(g*h\) denotes the convolution of g and h.

By the Laplace transform method, we obtain, with \({\hat{g}} (z)\) denoting the Laplace transform of g(t),A use of the inverse Laplace transform shows at \(t=t_{n}\)where \( \varGamma \) is defined by, see, e.g., Lubich et al. [23], with some \(\theta \in (\pi /2, \theta _{0})\),Now we shall consider the time discretization scheme of (1). To improve the accuracy near \(t=0\), we follow the approach in Lubich et al. [23] to correct the values of the first step in the time discretization.

Let \(V^{n}\approx V_{h}(t_{n})\) be the approximation of \(V_{h} (t_{n})\). We define the following time discretization scheme for approximating (1)where, with \(c_{0}=1/2\),Applying the discrete Laplace transform in both sides of (19), we obtainIt is easy to see with \({\tilde{V}} (\zeta ) = \sum _{j=0}^{\infty } V^{j} \zeta ^{j}\)which implies by (20) thatFurther, with \({\tilde{w}} (\zeta )\) given in (11), we setandBy the inverse discrete Laplace transform, it follows for \(n \ge 1\) and using the variable change \(\zeta = e^{-zk}\) thatwhere, see Lubich et al. [23], with \(\varGamma \) defined in (18),Below, we state our main result in this section whose proof will be provided subsequently.

To prove Theorem 1, the following two lemmas will be useful.

In the following lemma, with \(z_{k}\) defined in (21), we claim that \(z_{k}^{\alpha } \in \varSigma _{\theta _{0}}\) for some \(\theta _{0}\in (\frac{\pi }{2},\pi )\).

Now we turn to the proof of Theorem 1.

In this section, we introduce a third order time discretization scheme for solving (1) based on the weighted and shifted Grünwald-Letnikov difference operator introduced in Tian et al. [34].

Let us define the following weighted and shifted Grünwald-Letnikov difference operator \(\, _{L} D^{\alpha }_{k,p,q,r}\) to approximate the Riemann-Liouville fractional derivative operator \(\, _{0}^{R}D_{t}^{\alpha }\) bywhere p, q, r are integers and mutually nonequal, and \(B_{k,p}^{\alpha } \phi (t_{n})\) are defined by (6) andWhen \(p=0, \, q=-1, \, r=-2\), we obtainand we then arrive for \(n\ge 2\) atThus, we obtainwhereThe discrete Laplace transform of \(w_{j}^{(\alpha )}, j=0, 1, 2, \ldots \) is given byBelow, following the idea in Lemma 1, we consider the series expansion of the function \({\tilde{w}}(\zeta )\) in (41).

Next, we turn to the solution of (16) when \(f_{h}(t)\) is written aswhere \(g*h\) denotes the convolution of g and h.

An application of the Laplace transform to (16) with respect to the time variable t yieldsBy the inverse Laplace transform, the solution of (16) takes the following form at \(t=t_{n}\),Let \(V^{n}\approx V_{h}(t_{n}),n=0,1,2,\ldots ,N\) denote the approximate solution of the following time discretization scheme for solving (16), with \(V^{0}=0\),where \(w_{j}^{(\alpha )},j=0,1,2,\ldots \) are defined by (40) and the coefficients \(a_{1}, a_{2}, b_{1}, b_{2}\) satisfyandNow we come to the following main theorem in this section

To prove Theorem 2, we need the following lemmas.

In the following lemma, with \(z_{k}\) defined by (52), we will show that \(z_{k}^{\alpha } \in \varSigma _{\theta _{0}}\) for some \(\theta _{0} \in (\pi /2, \pi )\).

We are now ready to prove the main theorem of this section.

In order to prove the error analysis in the completely discrete schemes, we now recall the error estimates of the semidiscrete scheme as is developed in [13] for the problem with \(f=0\),Denote \(U^{n}= V^{n} + P_{h} u_{0}\) with \(V^{n}\) defined by (19) or (47)–(49), respectively. Using (65), we then have the following fully discrete error estimatewhere \(m=2\) or 3, respectively, for second order and third order time stepping schemes.

Since our main objective in this article is to derive higher order time stepping schemes, therefore, we may generalize our present results to include mass lumping scheme as discussed in [13].

This section is devoted to some generalization of the present methods to various other problems, which includes evolution equations with positive memory.

For instance, our generalizations include the following type of problems: Now, a use of Laplace transforms in (69) yieldswith some function h(z) depending on \(\alpha \). Hence, with \(\beta (z)=h(z)^{-1}\) we formally write the representation of solution as \({\hat{u}}=\beta (z) (z \beta (z) I+ A)^{-1} v =: {\hat{E}}_h(z)v.\) Here, for the problem (66), note that \(\beta (z)=z^{\alpha },\) for the problem (67), \(\beta (z)= z^\alpha /(1+z^{\alpha }),\) and for (68), \(\beta (z)=1/(1+\gamma z^\alpha )\).

Assume that one can properly choose \(\theta \) in \((\pi /2,\theta _0)\) with \(\theta _0 \in (\pi /2,\pi )\) such that \(z_k\beta (z_k)\in \varSigma _{\theta _0}\), where \(z_k\) is an approximation of \(z\in \varSigma _{\theta }.\) This is indeed possible in all given examples. With this, the resolvent estimate yieldswhere \(M_{\theta _0}=1/\sin (\pi -\theta _0)\).

With corresponding \({A}_h\) as semidiscrete approximation of A ( as in Sect. 2), the semidiscrete method becomes: find \({\bar{u}}_h(t)\in V_h\) such thatFollowing the arguments of [9, 23], one easily deduces property (70), when A is replaced by \(A_h\).

Therefore, our two time stepping methods can be appropriately applied and the desired estimates can be easily derived.

Other generalizations also include the fractional cable equations with mass lumping by Al-Maskari and Karaa [2] and references therein. Further, it may include Fokker-Plank spatial discretization as our analysis does not depend on selfadjointness of the operator A and for complete discrete scheme, we may use the proposed time stepping schemes.

Finally, the present schemes can be applied to the Eq. (1), when A is a fractional Laplacian like \(A= (-\varDelta )^{s},\;s\in (0,1)\), that is, with \( 0< \alpha , \beta <1\) and some positive constant \(\delta >0\),An appropriate modification of the arguments in [1], we obtain the semidiscrete error estimate as:with \(\epsilon > 0\) small. Therefore, when it is combined with our proposed time stepping schemes, the final error estimate reads as: with \(U^{n} \approx u_{h} (t_{n})\),

In this section, we present five numerical examples to show that the numerical results are consistent with the theoretical results obtained in this paper. The first three examples are solved by using the numerical method (19) for both homogeneous and inhomogeneous problems in one- and two-dimensional cases. The last two examples are computed by using the numerical method (47)–(49) for both homogeneous and inhomogeneous problems in one-dimensional case.

Let \(N_{h}\) be a positive integer. Let \(0=x_{0}< x_{1}< x_{2}< \cdots < x_{N_{h}} =1\) be the space partition and h the space step size. We shall use the piecewise linear finite element method to consider the space discretization.

Let \(0<t_{0}< t_{1}< \cdots < t_{N} =T\) be the time partition and k the time step size. To observe the convergence order of the numerical method, we first need to calculate the reference solution \(u_{ref}(t)\) at some fixed time T with very small step sizes \( h_{ref} = 2^{-6}\) and \( k_{ref} = 2^{-10}\).

By Theorem 1, we see that the numerical method (19) has the second order convergence. To see this convergence order, we shall calculate the approximate solution of u(T) at \(T=1\) with the space step size \(h=2^{-6}\) and the different time step sizes \( k =\kappa * k_{ref} \) with \(\kappa = [2^2, 2^3, 2^4,2^5, 2^6]\). In Table 2, we observe the convergence orders \(O(k^{2})\) of the corrected scheme (19), where the rows with (a) denote the errors and the experimentally determined convergence orders in the smooth data case and the rows with (b) denote the errors and the orders in the nonsmooth data case. For each \(\alpha \), we choose the average convergence order of the computed orders obtained by using the different time step sizes.

For the numerical method (19) with \(c_{0}=0\), that is, for the uncorrected scheme, we observe that, in Table 3, the experimentally determined convergence order is only O(k) with both smooth and nonsmooth data.

We use the same parameters as in the numerical simulations in Example 1. In Table 4, we observe that the experimentally determined convergence order of the corrected scheme (19) indeed is \(O(k^{2})\) for all \( 0< \alpha <1\) (Table 5).

Let \(N_{h}\) be a positive integer. Let \(0=x_{0}< x_{1}< x_{2}< \cdots < x_{N_{h}} =1\) and \(0=y_{0}< y_{1}< y_{2}< \cdots < y_{N_{h}} =1\) be the partition of \(\varOmega \). We divide \(\varOmega \) into some triangles with the same sizes and let h be the maximal length of the sides of the triangle. We shall use the piecewise linear finite element method to consider the space discretization on the triangulation of \(\varOmega \).

Let \(0<t_{0}< t_{1}< \cdots < t_{N} =T\) be the time partition and k the time step size. We shall use the very small space step size \( h_{ref} = 2^{-6}\) and the time step size \( k_{ref} = 2^{-10}\) to calculate the reference solution at time T.

We shall choose \(T=1\) in our simulation. We calculate the approximate solutions with the space step size \(h=2^{-6}\) and the time step sizes \( k =\kappa * k_{ref} \) with \(\kappa = [2^2, 2^3, 2^4,2^5, 2^6]\). In Table 6, the experimentally determined convergence orders \(O(k^{2})\) are observed as we expected.

In Table 7, we observe the experimentally determined convergence orders O(k) for the uncorrected scheme (19) with \(c_{0}=0\) as we expected.

In the next two examples, we shall consider the experimentally determined convergence orders of the numerical method (47)–(49).

For the uncorrected scheme (47)–(49), that is, \(a_{1}=a_{2}=b_{1}=b_{2}=0\) in (47)–(49), we observe that, in Table 9, the experimentally determined convergence order is only O(k) with both smooth and nonsmooth data.

In Table 11, the experimentally determined convergence order of the uncorrected scheme (47)–(49) with \(a_{1}=a_{2}=b_{1}=b_{2}=0\) has only convergence order O(k).

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