Stability and Positivity Preservation in Conventional Methods for Space-Fractional Diffusion Problems: Analysis and Algorithms

Abstract

The numerical solution of space-fractional diffusion problems is investigated focusing on stability and non-negativity issues. The extension of classical schemes is analyzed for the case of the spectral fractional Dirichlet Laplacian operator. For the spatial discretization, both finite differences and finite elements are used. The finite element case needs special care and is discussed in detail. Both spatial discretizations are combined with the matrix transformation method, leading to fractional powers of matrices in the discretized problems. In the time stepping, $\theta$-methods are utilized with $\theta =0,{\displaystyle \frac{1}{2}}$ and 1. In the analysis, it is pointed out that the stability condition in the case of $\theta =0$ depends on the fractional power $\alpha \in (0,1]$, which results in a weaker condition on the time discretization compared to the conventional diffusion. In this case, we also obtain non-negativity preservation. Also, unconditional stability is established for $\theta ={\displaystyle \frac{1}{2}}$ and $\theta =1$, where for the spatial discretization rather general conditions are posed. The results containing stability conditions are also confirmed in a series of numerical experiments. In the course of the corresponding algorithms, an efficient matrix power–vector product procedure is employed to keep simulation time at an affordable level. The associated computational algorithm is also described in detail.

1. Introduction

The investigation of the qualitative properties of solutions to partial differential equations (PDEs) is a classical problem. In time-dependent settings, for instance, the preservation of positivity is often essential from a modeling perspective. Moreover, certain qualitative properties—such as the maximum principle—serve as powerful tools in mathematical analysis. Accordingly, the preservation of these qualitative properties in the numerical solutions is also highly desirable. In the case of classical parabolic problems, using basic finite difference and finite element discretizations, some achievements are summarized in [1]. These results were extended to advanced spatial discretizations such as using special finite elements [2] or higher-order non-conforming (discontinuous) ones [3]. Also, the analysis was extended to certain non-linear problems. For a wider overview of the corresponding results, we refer to [4]. Regarding the time discretization, the Rosenbrock or, in general, implicit Runge–Kutta methods have optimal properties. For a general analysis, we refer to [5] and for a related practical case to [6].

Extending the study of qualitative properties to problems involving fractional diffusion is not straightforward. To begin with, several competing models exist for defining the spatial differential operator. See a survey on this issue in [7].

Since the fractional Laplacian is non-local, even this simple boundary condition has several interpretations. For a review on these, see [8], presenting different models mostly in the spatially one-dimensional case. Whenever the favor of the spectral (or operator type) approach was pointed out, also from the modeling point of view, many authors investigated the so-called integral fractional Laplacian [9] approach. This is also called the zero-extended or the Riesz fractional Laplacian [8].

Regarding this one, for the subdiffusive regime, non-negativity is preserved [10], while in the case of superdiffusion, this qualitative property is lost [11]. Investigating the continuous models of space-fractional diffusion, a unified abstract setting was developed to point out that the non-negativity of the initial data is preserved [12] if the spatial differential operator is negative definite with compact resolvent. This constitutes the continuous counterpart of our study.

Motivated by this result, we aim to find conditions, which imply the preservation of non-negativity for the numerical solution of space-fractional diffusion problems involving the spectral fractional Laplacian. Regarding the space discretization, we analyze both the case of finite differences and the finite elements. In both cases, we utilize the so-called matrix transformation method [13,14,15], where the case of finite elements needs a special care. Turning to the time discretization, where we investigate simple classical time steppings, we also analyze the stability of the corresponding scheme. This problem for the implicit Euler time stepping was studied in [13,15] and pointed out unconditional stability. We will point out that this property is independent of the fractional power. Owing to its high importance, the stability of the time discretization for space-fractional diffusion problems were investigated by many authors. The majority of studies, such as [16,17] or [18], were performed for the one-dimensional setup and distinguished more cases regarding the fractional power. In all cases, the unconditional stability of the Crank–Nicolson scheme was established. As an additional benefit of the matrix transformation approach, in the present framework, this analysis is carried out independently of the spatial dimension. Moreover, conditional stability is formulated in terms of the fractional power of the Laplacian operator. Finally, we give in detail the corresponding numerical algorithms, which ensure positivity preserving also at the discrete level. Beyond the theoretical results, we will also perform numerical experiments to validate our findings. For the computations, we employ a modern technique developed in [19] to avoid the costly explicit evaluation of matrix powers. Instead, matrix power–vector products are computed efficiently.

2. Mathematical Preliminaries

We investigate numerical algorithms for the solution of the following space-fractional diffusion model problem:

$$\left\{\begin{array}{c}{\partial }_{t}u(t,\mathbf{x})=-\mu ·{(-\Delta )}^{\alpha }u(t,\mathbf{x}),\mathbf{x}\in \Omega ,t>0\hfill \\ u(0,\mathbf{x})=u_0\left(\mathbf{x}\right),\mathbf{x}\in \Omega ,\hfill \end{array}\right.$$

(1)

where $\Omega \subset {\mathbb{R}}^d$ denotes the computational domain and $\mu \in {\mathbb{R}}^{+}$ is a diffusion coefficient. ${(-\Delta )}^{\alpha }$ denotes the fractional power of the negative Dirichlet Laplace operator which, for any $\alpha \in {\mathbb{R}}^{+}$, is defined using the spectral decomposition of $u\in C_0^{\infty }(\Omega )$ as follows:

$${(-\Delta )}^{\alpha }u(t,\mathbf{x})=\sum _{k=0}^{\infty }{u}_k{\lambda }_k^{\alpha }{\varphi }_k.$$

Here $-\Delta :L^2(\Omega )\to L^2(\Omega )$ denotes the negative Laplacian operator with homogeneous Dirichlet boundary conditions, ${\left\{{\varphi }_k\right\}}_{k\in \mathbb{N}}$ is the orthonormal system of its eigenfunctions with the corresponding eigenvalues ${\left\{{\lambda }_k\right\}}_{k\in \mathbb{N}}$, while $u_k=\underset{\Omega }{\int }u{\varphi }_k,k\in \mathbb{N}$ denotes the related Fourier coefficients.

In this way, homogeneous Dirichlet boundary conditions are imposed at the operator level. In a similar manner, homogeneous Neumann boundary conditions can also be incorporated. For further details, we refer to [7,13,20].

2.1. Spatial Discretization

Following the conventional way for the numerical solution of (1), we perform first the discretization of the Laplacian operator with $\alpha =1$. Also, for the simplicity, we will set $\mu =1$ in the forthcoming analysis. Note that in real-world problems, several additional terms—such as convection or reaction—may appear, and the magnitude of their associated coefficients can play a significant role.

According to the matrix transformation method, the case of general $\alpha$ can be obtained by taking the power $\alpha$ of the above classical discretization. Briefly, this method uses that in case of $A\approx -\Delta$, we also have $A^{\alpha }\approx {(-\Delta )}^{\alpha }$. We deal with the two classical choices in the following.

2.1.1. Finite Differences

If $A_{\mathrm{FD}}\in {\mathbb{R}}^{N\times N}$ denotes a finite difference discretization of $-\Delta$, by means of the matrix transformation method, we obtain the spatial semi-discretization

$${\partial }_t\underline{\mathbf{u}}(t,·)=-{\left(A_{\mathrm{FD}}\right)}^{\alpha }\underline{\mathbf{u}}(t,·),$$

(2)

for the original problem (1). Note that the matrix power ${\left(A_{\mathrm{FD}}\right)}^{\alpha }$ makes sense if $A_{\mathrm{FD}}$ is positive definite, which is satisfied in the case of classical finite differences. For this, see the forthcoming assumptions and the corresponding discussion. Also for the precise statement and the convergence results, we refer to [13].

2.1.2. Finite Elements

The case of the finite element spatial discretization needs a special care. For this, we use a basis ${\left\{b_j\right\}}_{j=1}^N$ in the standard energy space $H_0^1(\Omega )$. In case of piecewise polynomials, this ensures zero boundary data and interelement continuity. Accordingly, we are looking for the approximation

$$\sum _{j=1}^N{u}_j\left(t\right)·b_j\left(\mathbf{x}\right)\approx u(t,\mathbf{x}).$$

With this, the conventional diffusion problem in (1) with $\mu =1$ and $\alpha =1$ can be discretized as

$${\partial }_t\sum _{j=1}^N{u}_j\left(t\right)·b_j\left(\mathbf{x}\right)=\Delta \sum _{j=1}^N{u}_j\left(t\right)·b_j\left(\mathbf{x}\right)\iff \sum _{j=1}^N{\partial }_t{u}_j\left(t\right)·b_j\left(\mathbf{x}\right)=\sum _{j=1}^N{u}_j\left(t\right)·\Delta b_j\left(\mathbf{x}\right).$$

Testing this with all basis functions $b_k,k=1,2,\dots ,N$ and introducing the notation

$$\mathbf{u}=(u_1,u_2,\dots ,u_N):(0,T)\to {\mathbb{R}}^N$$

for the unknown time-dependent vector function, the equalities

$$\sum _{j=1}^N{\partial }_t{u}_j\left(t\right)·{\int }_{\Omega }{b}_j\left(\mathbf{x}\right)b_k\left(\mathbf{x}\right)\mathrm{d}\mathbf{x}=-\sum _{j=1}^N{u}_j\left(t\right)·{\int }_{\Omega }\nabla b_j\left(\mathbf{x}\right)·\nabla b_k\left(\mathbf{x}\right)\mathrm{d}\mathbf{x},k=1,2,\dots ,N$$

can gathered in the matrix–vector form

$$M{\partial }_t\mathbf{u}(t,·)=-K\mathbf{u}(t,·),$$

(3)

with the mass matrix $M\in {\mathbb{R}}^{N\times N}$ and the stiffness matrix $K\in {\mathbb{R}}^{N\times N}$, the entries of which are defined with

$$M[j,k]={\int }_{\Omega }{b}_j{b}_k\mathrm{and}K[j,k]={\int }_{\Omega }\nabla b_j·\nabla b_k,$$

respectively. In this way, (3) finally constitutes the semidiscretized problem for $\alpha =1$ in the case of the above finite element discretization, which is rewritten as

$${\partial }_t\mathbf{u}(t,·)=-M^{-1}K\mathbf{u}(t,·).$$

Accordingly, in this case, the matrix $-M^{-1}K$ should be considered as the discretized Laplacian. Hence, the semidiscrete form of (1) for a general fractional power $\alpha$ becomes

$${\partial }_t\mathbf{u}(t,·)=-{\left(M^{-1}K\right)}^{\alpha }\mathbf{u}(t,·).$$

(4)

For further details, including the convergence theory of this approach, we refer to [14].

Since K and M are symmetric positive definite matrices arising from a finite element discretization of an elliptic operator, the generalized eigenvalue problem $K\mathbf{v}=\lambda M\mathbf{v}$ gives non-negative real eigenvalues and M-orthogonal eigenvectors. Consequently, $M^{-1}K$ is diagonalizable with a non-negative real spectrum. For this statement, see [21]. Using this fact, the matrix power in (4) makes sense.

2.2. Time Discretization

For a unified treatment, we use the notation A, which refers to $A_{\mathrm{FD}}$ and $M^{-1}K$ in the case of using finite differences and finite elements, respectively.

We investigate the time discretization of (2) or (4), which is, in other words, the full discretization of (1). For this, we consider the weighted average of the explicit and implicit one-step schemes, leading to the so-called $\theta$ schemes, expressed as follows:

$${\displaystyle \frac{{\mathbf{u}}^{n+1}-{\mathbf{u}}^n}{\delta }}=-{\left(A\right)}^{\alpha }{\mathbf{u}}^n·(1-\theta )+-{\left(A\right)}^{\alpha }{\mathbf{u}}^{n+1}·\theta ,$$

(5)

where $\delta$ yields the time step and $\theta \in [0,1]$ is a positive parameter. For $\theta =0$, we get the explicit Euler scheme; otherwise, it is an implicit one. The choice $\theta ={\displaystyle \frac{1}{2}}$ leads to the Crank–Nicolson scheme.

2.3. Stability and Non-Negativity Preservation

The following classical definition is the cornerstone of the analysis for the non-negativity preservation in the case of $\alpha =1$.

Definition 1.

The matrix $B\in {\mathbb{R}}^{N\times N}$ is said to be an M matrix if

• All its off-diagonal elements are non-positive.
• There exists a vector $\mathbf{g}\in {\mathbb{R}}^N>0$ such that $B\mathbf{g}>0$,

where the inequalities are understood componentwise.

Any M matrix is invertible and has a componentwise non-negative inverse. Moreover, the norm of the inverse can be estimated using the vector $\mathbf{g}$. At the same time, we have to use another somewhat forgotten notion for the present analysis.

Definition 2.

The matrix $B\in {\mathbb{R}}^{N\times N}$ is called a Stieltjes matrix if

• B is a real symmetric positive definite matrix.
• All its off-diagonal elements are non-positive.

Note that a matrix $B\in {\mathbb{R}}^{N\times N}$ is called (weakly) diagonally dominant if

$$\begin{gathered} \sum _{\begin{array}{c}k=1 \\ k\ne j\end{array}}^N|B_{jk}|\le |B_{jj}|\mathrm{for}\mathrm{all}j=1,\dots ,N, \end{gathered}$$

where $B_{jk}$ are the elements of the matrix B. Regarding this notion, we will use the following key properties in the forthcoming analysis.

Theorem 1.

Let B be a Stieltjes matrix and $\alpha \in (0,1]$. In this case,

• B is an M matrix such that it has a non-negative inverse.
• $B^{\alpha }$ is also a Stieltjes matrix.

For the details, we refer to [22].

3. Results

3.1. Analysis for Stability and Positivity Preserving

Regarding the spatial discretization, we formulate only some general assumptions in the following.

(i)

A is symmetric positive definite.

(ii)

A has positive diagonal elements and non-positive off-diagonals elements.

(iii)

A is (weakly) diagonally dominant.

(iv)

The diagonal of A is $d·I$ with $d\ge 1$, where I is the identity matrix.

Since they are used throughout in the forthcoming analysis, we discuss them in detail.

• We first note that any meaningful approximation A of $-\Delta$ should satisfy Assumption (i) as the original operator $-\Delta$ equipped with Dirichlet boundary conditions is a self-adjoint positive operator.
• The positivity of A implies that the diagonal elements of A become positive. Also, in all conventional finite difference discretizations, this is satisfied. For a related statement, see [23]. The non-positivity in Assumption (ii) for the stiffness matrix K will be satisfied using some geometric conditions regarding the finite element mesh, see, e.g., [24]. This assumption should be satisfied if the non-negativity is preserved in the case of $\alpha =1$. Note that this condition can fail by using higher-order spatial approximation.
• According to (ii), for Assumption (iii), we have only to verify the non-negativity of the sum of rows in A. Note also that any consistent scheme applied to the constant function gives a zero-sum in the matrix rows corresponding to the grid points which have interior neighboring points. Also, for the rest of the rows, some negative components will cancel out, turning the row positive and satisfying Assumption (iii).
• Finally, to satisfy Assumption (iv), it is satisfactory to have a uniform spatial discretization.

We will investigate separately the different classical time discretizations. To avoid computing with full matrices, we also investigate the explicit Euler discretization (case of $\theta =0$ in (5)), given as

$${\mathbf{u}}^{n+1}={\mathbf{u}}^n-\delta A^{\alpha }{\mathbf{u}}^n=(I-\delta A^{\alpha }){\mathbf{u}}^n.$$

(6)

Regarding this, we pose an extra assumption.

(v)

The inequality $1-\delta ·d\ge 0$ is satisfied for time step $\delta$ in (6).

• This condition for explicit schemes is also a natural one as follows: since d depends on the spatial discretization parameter, this can be regarded as a stability condition.

For example, applying the standard five-point second-order finite difference scheme to approximate the Dirichlet Laplacian on a uniform square grid with parameter h, we obtain $d={\displaystyle \frac{4}{h^2}}$. In this way (v), is equivalent with $\delta \le {\displaystyle \frac{h^2}{4}}$, yielding the standard stability condition for $\delta$ in two space dimensions.

Theorem 2.

Assume that conditions (i)–(v) are satisfied for the time step δ in (6). Then the explicit time stepping in (6) is stable and preserves non-negativity for any $\alpha \in (0,1].$

Proof. 

Using the condition in (v), where d denotes the diagonal elements of A, we can equivalently rewrite it as

$$A=d·(I-B),$$

(7)

where $B\in {\mathbb{R}}^{N\times N}$ with $B_{jk}=-{\displaystyle \frac{1}{d}}{A}_{jk}\ge 0$ for $k\ne j$ and $B_{jj}=0$. Once again, lower indices here denote the related matrix entries.

On the other hand, using Assumption (iii),

$$\sum _{k=1}^N|B_{jk}|={\displaystyle \frac{1}{d}}\sum _{k\ne j}^N\left|A_{jk}\right|\le 1\mathrm{for}\mathrm{all}j=1,\dots ,N,$$

such that ${\parallel B\parallel }_{\infty }\le 1$. In this way, using (7), the binomial series expansion for $\alpha \in (0,1]$ gives that

$$A^{\alpha }=d^{\alpha }·{(I-B)}^{\alpha }=d^{\alpha }·\sum _{k=0}^{\infty }{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)B^k=d^{\alpha }·\left(I+\sum _{k=1}^{\infty }{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)B^k\right).$$

(8)

Here ${(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)<0$ for all $k\ge 1$, while B is element-wise non-negative, and therefore, the sum in the second term is non-positive. In this way, the off-diagonal elements of $I-\delta A^{\alpha }$ are all non-negative.

Also, using Assumption (iv) again, in (8), the diagonal elements of $A^{\alpha }$ satisfy

$$A_{jj}^{\alpha }\le d^{\alpha }\le d.$$

In this way, using Assumption (v), the diagonal elements of $I-\delta A^{\alpha }$ can be estimated as

$$1-\delta A_{jj}^{\alpha }\ge 1-\delta ·d\ge 0,$$

such that, summarized, $I-\delta A^{\alpha }$ is element-wise non-negative, which implies the non-negativity preservation as stated in the theorem. □

Conventionally, the matrix A is given by $A={\displaystyle \frac{1}{h^2}}{A}_0$, where $A_0\in {\mathbb{R}}^{N\times N}$ and h is a grid parameter, usually the minimal grid size in a finite element or finite element discretization. In this framework, conditions for stability or the preservation of non-negativity appear as an upper bound $c_{*}$ for the ratio ${\displaystyle \frac{\delta }{h^2}}$. In practice, regarding the time step $\delta$, this can be formulated as $\delta \le c_{*}·h^2$, which is frequently the bottleneck in developing fast numerical simulations. We point out that this condition can be somewhat mitigated in the case of fractional order diffusion provided that $\alpha <1$.

Theorem 3.

Assume that, for $\alpha =1$, we have the stability condition $\delta \le 2c_{*}·h^2$ with some $c_{*}\in {\mathbb{R}}^{+}$ in (6). Then, for any $\alpha \in (0,1]$, the condition $\delta \le 2c_{*}^{\alpha }·h^{2\alpha }$ in (6) is sufficient for the same purpose.

Proof. 

The decomposition of the matrix A for $\alpha =1$ gives that

$$I-\delta A=I-\delta {\displaystyle \frac{A_0}{h^2}}=I-{\displaystyle \frac{\delta }{h^2}}{A}_0,$$

(9)

and using the matrix transformation method, we get

$$I-\delta {\left({\displaystyle \frac{A_0}{h^2}}\right)}^{\alpha }=I-{\displaystyle \frac{\delta }{h^{2\alpha }}}{\left(A_0\right)}^{\alpha }.$$

(10)

Since $A_0$ is symmetric with positive eigenvalues, the stability of (6) is equivalent with

$${\lambda }_{min}\left\{I-\delta ·{\left({\displaystyle \frac{A_0}{h^2}}\right)}^{\alpha }\right\}\ge -1,$$

which can be rewritten as

$${\lambda }_{max}\left\{\delta ·{\left({\displaystyle \frac{A_0}{h^2}}\right)}^{\alpha }\right\}={\lambda }_{max}\left\{\delta ·{\left({\displaystyle \frac{A_0}{h^2}}\right)}^{\alpha }\right\}\le 2.$$

This obviously implies the sufficient upper bound

$$\delta \le 2·{\displaystyle \frac{1}{{\lambda }_{max}\left\{A_0^{\alpha }\right\}}}·h^{2\alpha }=2·{\left({\displaystyle \frac{1}{{\lambda }_{max}\left\{A_0\right\}}}\right)}^{\alpha }·h^{2\alpha },$$

(11)

which for $\alpha =1$ gives

$$\delta \le 2·{\displaystyle \frac{1}{{\lambda }_{max}\left\{A_0\right\}}}·h^2.$$

In this way, for any $c_{*}$, which can be used in the stability condition $\delta \le 2c_{*}·h^2$, we have

$$c_{*}\le {\displaystyle \frac{1}{{\lambda }_{max}\left\{A_0\right\}}}.$$

Therefore, in the case of any $\alpha \in (0,1]$, the condition $\delta \le 2c_{*}^{\alpha }·h^{2\alpha }$ implies that

$$\delta \le 2c_{*}^{\alpha }·h^{2\alpha }\le 2·{\left({\displaystyle \frac{1}{{\lambda }_{max}\left\{A_0\right\}}}\right)}^{\alpha }·h^{2\alpha },$$

which, according to (11), is really a sufficient condition for stability. □

Remark 1.

In the one-dimensional case, applying the second-order central finite difference approximation of the Laplacian on a uniform grid of size h, we have the stability condition $\delta \le {\displaystyle \frac{1}{2}}{h}^2=2·{\displaystyle \frac{1}{4}}·h^2$ in (6) with $\alpha =1$. According to Theorem 3, in the case of a general $\alpha \in (0,1]$, the corresponding stability condition becomes $\delta \le 2·{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha }·h^{2\alpha }.$

In the two-dimensional case, applying the standard symmetric five-point finite difference stencil on a uniform square grid of size h, we have the stability condition $\delta \le {\displaystyle \frac{1}{4}}{h}^2=2·{\displaystyle \frac{1}{8}}·h^2$ in (6) with $\alpha =1$. According to Theorem 3, in the case of a general $\alpha \in (0,1]$, the corresponding stability condition becomes $\delta \le 2·{\left({\displaystyle \frac{1}{8}}\right)}^{\alpha }·h^{2\alpha }.$

In the two-dimensional case, applying bilinear finite elements on a uniform square grid of size h, we have again the stability condition $\delta \le {\displaystyle \frac{1}{2}}{h}^2=2·{\displaystyle \frac{1}{4}}·h^2$ in (6) with $\alpha =1$. According to Theorem 3, in the case of a general $\alpha \in (0,1]$, the corresponding stability condition becomes $\delta \le 2·{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha }·h^{2\alpha }.$

Remark 2.

For taking $h\ll 1$ and $\alpha <1$, the stability constraint $\delta \le 2c_{*}^{\alpha }·h^{2\alpha }$ for the time step may be significantly less restrictive compared to $\delta \le 2c_{*}·h^2$.

We turn now to the case of implicit time stepping. The implicit Euler discretization, i.e., the case of $\theta =1$ in (5), is given as follows:

$${\mathbf{u}}^{n+1}={\mathbf{u}}^n-\delta A^{\alpha }{\mathbf{u}}^{n+1}.$$

(12)

Theorem 4.

Assume that the conditions (i) and (ii) are satisfied. Then implicit time stepping is stable for any $\alpha \in (0,1]$ and preserves non-negativity.

Proof. 

Using the implicit scheme in (12), we have

$${\mathbf{u}}^{n+1}={[I+\delta ·A^{\alpha }]}^{-1}{\mathbf{u}}^n,$$

(13)

where $A^{\alpha }$ is a Stieltjes matrix by Theorem 1. The same applies to $I+\delta ·A^{\alpha }$, such that the inverse in (13) contains only non-negative elements. Therefore, the non-negativity of the numerical solution holds unconditionally. □

Since the above time discretizations result only in a first-order convergence, we also investigate the Crank–Nicolson scheme

$${\mathbf{u}}^{n+1}={\mathbf{u}}^n+{\displaystyle \frac{\delta }{2}}{A}^{\alpha }({\mathbf{u}}^n+{\mathbf{u}}^{n+1}),$$

which can be rephrased as

$${\mathbf{u}}^{n+1}={\left(I+{\displaystyle \frac{\delta }{2}}{A}^{\alpha }\right)}^{-1}\left(I-{\displaystyle \frac{\delta }{2}}{A}^{\alpha }\right){\mathbf{u}}^n.$$

(14)

In this case, however, we have different conditions for the stability and the non-negativity preservation. Regarding this, along with Theorem 3, we state the following.

Theorem 5.

The Crank–Nicolson scheme in (14) is unconditionally stable for any $\alpha \in (0,1].$

Proof. 

We introduce the notation $W={\displaystyle \frac{\delta }{2}}{A}^{\alpha }$, where, according to assumption (i), W is also a positive definite symmetric matrix. Then the complete set of its eigenvectors and the corresponding non-negative eigenvalues are denoted by $\{{\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_N\}$ and $\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N\}$, respectively.

We first note that the matrix ${\left(I+{\displaystyle \frac{\delta }{2}}{A}^{\alpha }\right)}^{-1}\left(I-{\displaystyle \frac{\delta }{2}}{A}^{\alpha }\right)={(I+W)}^{-1}(I-W)$ is also symmetric. To verify this, we compute

$${\left[{(I+W)}^{-1}(I-W)\right]}^T={(I-W)}^T{\left({(I+W)}^T\right)}^{-1}=(I-W){(I+W)}^{-1},$$

and observe that multiplying it from the left with $I+W$ gives

$$(I+W)(I-W){(I+W)}^{-1}=(I-W)(I+W){(I+W)}^{-1}=I-W,$$

which is the same as multiplying ${(I+W)}^{-1}(I-W)$ from the left with $I+W$. Since $I+W$ is invertible, this verifies the equality

$${\left[{(I+W)}^{-1}(I-W)\right]}^T={(I+W)}^{-1}(I-W).$$

In this way, for the unconditional stability of the scheme in (14), it is sufficient to verify that all eigenvalues of ${(I+W)}^{-1}(I-W)$ lie in the interval $[-1,1]$. For this, we observe that any eigenvector ${\mathbf{w}}_j$ of W is also an eigenvector of both $I-W$ and ${(I+W)}^{-1}$. In this way, by the completeness of $\{{\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_N\}$, this system also constitutes the eigenvectors of ${(I+W)}^{-1}(I-W)$. Moreover, we can give the corresponding eigenvalues as

$$\left\{{\displaystyle \frac{1-{\lambda }_1}{1+{\lambda }_1}},{\displaystyle \frac{1-{\lambda }_2}{1+{\lambda }_2}},\dots ,{\displaystyle \frac{1-{\lambda }_N}{1+{\lambda }_N}}\right\},$$

such that the non-negativity of ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N$ implies that ${\displaystyle \frac{1-{\lambda }_j}{1+{\lambda }_j}}\in [-1,1],j=1,2,\dots ,N$, which completes the proof. □

In Theorem 2, we verified that the time step $\delta$, ensuring the stability and non-negativity preservation for $\alpha =1$, can also be used for any $\alpha \in (0,1]$. At the same time, according to Theorem 3, in the case of $\alpha <1$, we could also use larger time steps compared with the case of $\alpha =1$. In practical cases, it can speed up the simulations. We present an experimental analysis addressing this issue using the explicit Euler time-stepping method.

3.2. Algorithmic Details

A main compound of an efficient explicit time discretization is discussed first. Then we turn to the treatment of the finite element spatial discretization.

3.2.1. Computation of Matrix–Vector Products

For the computation of matrix power–vector products $A^{\alpha }\mathbf{v}$, where $A\in {\mathbb{R}}^{n\times n}$ is a symmetric positive definite sparse matrix, $\mathbf{v}\in {\mathbb{R}}^n$ is an arbitrary vector, and $\alpha \in (0,1]$, we apply the Matlab codes described in [19]. Using this technique, we can avoid the calculation of the matrix power $A^{\alpha }$, which is a large and full matrix. Within the framework of the above technique, only sparse matrix–vector products need to be computed, resulting in an efficient implementation.

Let ${\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_n$ are the eigenvectors and ${\lambda }_1\le {\lambda }_2\le \dots \le {\lambda }_n$ the corresponding eigenvalues of A. Then $A^{\alpha }$ has the same eigenvectors, while the corresponding eigenvalues are ${\lambda }_1^{\alpha }\le {\lambda }_2^{\alpha }\le \dots \le {\lambda }_n^{\alpha }$. For any vector,

$$\mathbf{v}=a_1{\mathbf{x}}_1+a_2{\mathbf{x}}_2+\dots +a_n{\mathbf{x}}_n\in {\mathbb{R}}^n.$$

The matrix–vector product is given as follows:

$$A^{\alpha }\mathbf{v}=a_1{\lambda }_1^{\alpha }{\mathbf{x}}_1+a_2{\lambda }_2^{\alpha }{\mathbf{x}}_2+\dots +a_n{\lambda }_n^{\alpha }{\mathbf{x}}_n,$$

(15)

which has to be approximated. To reduce the computational complexity, we compute only the following few eigenvectors: ${\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_{k_1}$ and ${\mathbf{x}}_{n-k_2+1},{\mathbf{x}}_{n-k_2+2},\dots ,{\mathbf{x}}_n$, along with the corresponding eigenvalues. This can be performed rather quickly using the sparse eigensolvers of Matlab and can be used in the course of the entire simulation process. The approximation has then the following two main steps:

• We first compute the component

  $${\mathbf{v}}_1=a_1{\mathbf{x}}_1+a_2{\mathbf{x}}_2+\cdots +a_{k_1}{\mathbf{x}}_{k_1}+a_{n-k_2+1}{\mathbf{x}}_{n-k_2+1}+\cdots +a_n{\mathbf{x}}_n.$$
  For this part, we can easily apply (15) to get

  $$A^{\alpha }{\mathbf{v}}_1=a_1{\lambda }_1^{\alpha }{\mathbf{x}}_1+\cdots +a_{k_1}{\lambda }_{k_1}^{\alpha }{\mathbf{x}}_{k_1}+a_{n-k_2+1}{\lambda }_{n-k_2+1}^{\alpha }{\mathbf{x}}_{n-k_2+1}+\cdots +a_n{\lambda }_n^{\alpha }{\mathbf{x}}_n.$$

  (16)
• For the rest ${\mathbf{v}}_2=\mathbf{v}-{\mathbf{v}}_1$, we apply a truncated Taylor’s approximation

$$A^{\alpha }{\mathbf{v}}_2\approx \left({\displaystyle \frac{\sigma \left(A\right)}{2}}{)}^{\alpha }\sum _{k=1}^K\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)\right({\displaystyle \frac{2A}{\sigma \left(A\right)}}-I{)}^k{\mathbf{v}}_2,$$

(17)

where $\sigma$ denotes the spectral radius and $\sigma ({\displaystyle \frac{2A}{\sigma \left(A\right)}}-I)\le 1$, such that the above binomial series is convergent.

Finally, we obtain the approximation $A^{\alpha }\mathbf{v}\approx A^{\alpha }{\mathbf{v}}_1+A^{\alpha }{\mathbf{v}}_2.$

3.2.2. Efficient Spatial Discretizations

In the case of finite difference discretizations, the matrix $A_{\mathrm{FD}}$ in (2) can be based on any conventional finite difference approximation of $-\Delta$. This becomes sparse such that the products $A^{\alpha }\mathbf{v}$ in Section 3.2.1 can be computed using only sparse matrix–vector products.

At the same time, in the case of finite element discretizations, the matrix $M^{-1}K$ can also be dense for sparse matrices M and K. A remedy for this problem is offered by applying mass lumping. Accordingly, we use $\tilde{M}\approx M$, where $\tilde{M}$ is the mass-lumped diagonal matrix with

$$\tilde{M}[j,j]=\sum _{k=1}^{N}M[j,k].$$

(18)

In this case, ${\tilde{M}}^{-1}K$ remains sparse.

For an overview of this technique with error analysis, we refer to [25].

3.2.3. Summary of the Computing Algorithm

Based on the above compounds, we summarize the steps of the explicit method for the numerical solution of (1).

1.

Compute matrix A for the spatial discretization.

• In the case of finite differences, this is a conventional finite difference approximation of $-\Delta$ with Dirichlet boundary data.
• In the case of finite elements, compute the stiffness matrix K according to Assumptions (i)–(iv).
  For example, one can choose an acute triangular or tetrahedral mesh with first-order elements or a rectangular mesh with bilinear elements.

Then compute the mass matrix M and apply the mass lumping in (18).

Finally, $A=-{\tilde{M}}^{-1}K$.

2.

Apply the matrix power–vector product in Section 3.2.1 to perform the time step

$${\mathbf{u}}^{n+1}={\mathbf{u}}^n-\delta A^{\alpha }{\mathbf{u}}^n.$$

(19)

3.3. Numerical Experiments

The computations in the forthcoming examples will confirm our theoretical expectations in Theorems 2 and 3.

Example 1.

We consider the one-dimansional space-fractional diffusion problem as follows:

$$\left\{\begin{array}{c}{\partial }_{t}u(t,x)=-{(-\Delta )}^{\alpha }u(t,x),x\in (0,\pi /2),t\in (0,0.1)\hfill \\ u(0,x)=sin\left(2x\right),x\in (0,\pi /2)\hfill \\ u(t,0)=u(t,\pi /2)=0,t\in (0,0.1),\hfill \end{array}\right.$$

(20)

where the analytic solution is given by $u(t,x)=e^{-4^{\alpha }t}sin2x$.

In this experiment, we applied different exponents $\alpha \in \{0.4,0.6,0.8\}$ in the matrix (or operator) power. For the spatial discretization, we apply a uniform grid with n subintervals such that we have the grid size $h={\displaystyle \frac{\pi }{2n}}$. Performing the conventional second-order finite difference approximation of $-\Delta$, we get A in (19).

In the decomposition in (16), we have used $k_1=k_2=20$ as the number of eigenvectors of A, while the first $K=1000$ terms were computed in the Taylor expansion in (17). For more details, see Section 3.2.1.

For the time steps, we use $\delta ={10}^{-4}$ in each case. The computational errors in the discrete $L_2$-norm and the corresponding convergence rates are given in Table 1.

Table 1. Computational errors and convergence results with respect to the spatial parameter h for the matrix transformation method applied to (20). In all cases, the first $K=1000$ terms in the Taylor expansion (17) were computed and the parameters $k_1=k_2=20$ were used in (16). $r_{0.4}$, $r_{0.6}$, and $r_{0.8}$ denote the convergence order with respect to the space parameter h in the case of $\alpha =0.4$, $\alpha =0.6$, and $\alpha =0.8$, respectively.

${\mathit{L}}_2$–Error
$\mathit{\delta }$	$\mathit{h}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{4}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{6}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{8}$	${\mathit{r}}_{\mathbf{0}.\mathbf{4}}$	${\mathit{r}}_{\mathbf{0}.\mathbf{6}}$	${\mathit{r}}_{\mathbf{0}.\mathbf{8}}$
${10}^{-4}$	0.2618	$1.3\times {10}^{-3}$	$2.4\times {10}^{-3}$	$3.8\times {10}^{-3}$	-	-	-
${10}^{-4}$	0.1309	$4.143\times {10}^{-4}$	$6.97\times {10}^{-4}$	$1.1\times {10}^{-3}$	1.6	1.7	1.7
${10}^{-4}$	0.0654	$1.221\times {10}^{-4}$	$2.158\times {10}^{-4}$	$3.235\times {10}^{-4}$	1.7	1.6	1.7
${10}^{-4}$	0.0327	$3.817\times {10}^{-5}$	$7.192\times {10}^{-5}$	$8.987\times {10}^{-5}$	1.6	1.5	1.8
${10}^{-4}$	0.0164	$1.272\times {10}^{-5}$	$2.258\times {10}^{-5}$	$2.248\times {10}^{-5}$	1.5	1.6	1.9

Note that for a fixed $\delta$, this results in a non-negligible error, such that the rates $r_{0.4}$, $r_{0.6}$, and $r_{0.8}$ do not show a real spatial convergence rate. At the same time, we can point out here that taking smaller powers introduces extra error in the computations.

Also, the numerical solution was positive in the interior of the domain. Since we have computed with a time step satisfying conditions (i)–(iv), this confirms the statement in Theorem 2.

We also investigate the stability conditions by comparing the maximum time step ${\delta }_{\exp }$ in the experiments and the maximal time step ${\delta }_{\mathrm{theory}}=2·{\displaystyle \frac{h^{2\alpha }}{4^{\alpha }}}$, ensuring stability according to Theorem 3 and Remark 1. To validate the reliability of this bound, we have performed the computations for more fixed spatial discretization parameters h. The corresponding results for the numerical solution of (20) are recorded in Table 2. These confirm the accuracy of the stability condition stated in Theorem 3.

Table 2. Comparison of experimental stability results in the numerical solution of (20) and the related theoretical stability results in Theorem 3. The maximal time steps leading to a stable simulation are shown for different values of h and fractional powers $\alpha$.

	$\mathit{\alpha }=0.4$			$\mathit{\alpha }=0.6$			$\mathit{\alpha }=0.8$
h	0.0785	0.0392	0.0314	0.0785	0.0392	0.0314	0.0785	0.0392
${\delta }_{\exp }$	0.145	0.0856	0.0714	0.0405	0.0175	0.0132	0.0109	0.0032
${\delta }_{\mathrm{theory}}$	0.150	0.0861	0.0721	0.0411	0.0179	0.0137	0.0113	0.0037

Example 2.

We consider the following two-dimensional space-fractional diffusion problem:

$$\left\{\begin{array}{c}{\partial }_{t}u(t,x)=-{(-\Delta )}^{\alpha }u(t,x,y),(x,y)\in \Omega ,t\in (0,0.1)\hfill \\ u(t,x,y)=0,(x,y)\in \partial \Omega ,t\in (0,0.1)\hfill \\ u(0,x,y)=sinxsin2y+sin2xsiny,(x,y)\in \Omega ,\hfill \end{array}\right.$$

(21)

where $\Omega =(0,\pi )\times (0,\pi )$ and the analytic solution is given by

$$u(t,x,y)=e^{-5^{\alpha }t}(sinxsin2y+sin2xsiny).$$

In this two-dimensional numerical experiment, we verify the global convergence order. For this, the time steps were multiplied with ${\displaystyle \frac{1}{4}}$ and the spatial parameter h with ${\displaystyle \frac{1}{2}}$ in the consecutive computations. Again, we use uniform square grids. The computational errors in the discrete $L_2$-norm are given in Table 3. These confirm the convergence order O($\delta$)+O($h^2$) stated in [13]. This is a favorable property of the following matrix transformation method: the spatial convergence order does not depend on the fractional power $\alpha$. Also, according to Theorem 2, in each experiment, we obtained a non-negative numerical solution.

Table 3. Computational errors for the matrix transformation method applied to (21). In all cases, the first $K=1000$ terms in the Taylor expansion (17) were computed and the parameters $k_1=k_2=20$ were used in (16).

${\mathit{L}}_2$–Error
$\mathit{\delta }$	$\mathit{h}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{4}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{6}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{8}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{9}$
$0.02$	0.5236	$7.49\times {10}^{-2}$	$1\times {10}^{-1}$	$1.29\times {10}^{-1}$	$1.45\times {10}^{-1}$
$0.02·{\displaystyle \frac{1}{4}}$	0.2618	$1.86\times {10}^{-2}$	$2.48\times {10}^{-2}$	$3.18\times {10}^{-2}$	$3.54\times {10}^{-2}$
$0.02·{\left({\displaystyle \frac{1}{4}}\right)}^2$	0.1309	$4.6\times {10}^{-3}$	$6.2\times {10}^{-3}$	$7.9\times {10}^{-3}$	$8.8\times {10}^{-3}$
$0.02·{\left({\displaystyle \frac{1}{4}}\right)}^3$	0.0654	$1.1\times {10}^{-3}$	$1.5\times {10}^{-3}$	$2.0\times {10}^{-3}$	$2.2\times {10}^{-3}$
$0.02·{\left({\displaystyle \frac{1}{4}}\right)}^4$	0.0327	$2.78\times {10}^{-4}$	$3.81\times {10}^{-4}$	$5.2\times {10}^{-4}$	$5.56\times {10}^{-4}$

Example 3.

We consider the following 2D space-fractional diffusion problem:

$$\left\{\begin{array}{c}{\partial }_{t}u(t,x)=-{(-\Delta )}^{\alpha }u(t,x,y),(x,y)\in \Omega ,t\in (0,0.1)\hfill \\ u(t,x,y)=0,(x,y)\in \partial \Omega ,t\in (0,0.1)\hfill \\ u(0,x,y)=sinxsiny,(x,y)\in \Omega ,\hfill \end{array}\right.$$

(22)

where $\Omega =(0,\pi )\times (0,\pi )$ and the analytic solution is given by $u(t,x,y)=e^{-2^{\alpha }t}(sinxsiny)$.

In this case, we relate the computational results in case of finite difference and finite element spatial discretization.

• As in Example 2, we have first used a uniform square mesh and applied again the conventional five-point finite difference approximation of the Laplacian.
• In the second series of experiments, on the same square mesh, bilinear finite element space is constructed. Computing the related mass matrix M and the stiffness matrix K is performed as described in Section 2.1.1. Note that, here, the assumptions in (i)–(iv) are again satisfied.

The computational errors and the corresponding convergence rates in the discrete $L_2$-norm applying the finite element-based algorithm are given in Table 4. For also verifying the convergence rate $O\left(\delta \right)+O\left(h^2\right)$, we refined the time step $\delta$ and the spatial parameter h with a factor ${\displaystyle \frac{1}{4}}$ and ${\displaystyle \frac{1}{2}}$, respectively. The ratio of errors in the last refinement is also displayed. Here, ratio 4 would correspond to the desired convergence rate, which is closely approximated. To obtain this result, we had to refine the computation of matrix power–vector products in Section 3.2.1. We made use of the first and last 25-25 eigenvalues and eigenvectors and computed 1500 terms in the Taylor approximation.

Table 4. Computational errors and convergence results with respect to the spatial parameter h for the matrix transformation method applied to (20). In all cases, the first $K=1500$ terms in the Taylor expansion (17) were computed and the parameters $k_1=k_2=25$ were used in (16). Last row: the ratio of the final two errors.

${\mathit{L}}_2$–Error
	$\mathit{\delta }$	$\mathit{h}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{2}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{5}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{6}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{8}$
	0.02	0.62832	$3.1\times {10}^{-2}$	$3.72\times {10}^{-2}$	$3.95\times {10}^{-2}$	$4.5\times {10}^{-2}$
	$0.02·{\displaystyle \frac{1}{4}}$	0.3141	$7.99\times {10}^{-3}$	$1.03\times {10}^{-2}$	$1.12\times {10}^{-2}$	$1.33\times {10}^{-2}$
	$0.02·{\left({\displaystyle \frac{1}{4}}\right)}^2$	0.157	$2.04\times {10}^{-3}$	$2.7\times {10}^{-3}$	$2.96\times {10}^{-3}$	$3.53\times {10}^{-3}$
	$0.02·{\left({\displaystyle \frac{1}{4}}\right)}^3$	0.0785	$5.15\times {10}^{-4}$	$6.89\times {10}^{-4}$	$7.57\times {10}^{-4}$	$9.07\times {10}^{-4}$
ratio			3.96	3.92	3.91	3.89

We have again compared the theoretical predictions in Theorem 3 with the experimental bound for stability for different exponents $\alpha$ in case of a few spatial discretization parameters h. According to Remark 1, in the case of the symmetric five-point finite difference scheme, we apply ${\delta }_{\mathrm{theory}}=2·{\left({\displaystyle \frac{1}{8}}\right)}^{\alpha }·h^{2\alpha }$, while in the case of bilinear finite elements, we use the stability bound ${\delta }_{\mathrm{theory}}=2·{\left({\displaystyle \frac{1}{4}}\right)}^{\alpha }·h^{2\alpha }$.

The results shown in Table 5 and Table 6 confirm the above theoretical predictions arising from Theorem 3. This also shows the advance of the finite element approach as follows: for the same spatial discretization, we can perform larger time steps. This is also valid in the case of the fractional operators.

Table 5. Comparison of experimental and predicted stability results by using the standard five-point finite difference approximation on a uniform grid in Example 2. The maximal time steps ensuring stability are related for different values of h and fractional powers $\alpha$. ${\delta }_{\mathrm{theory}}$: maximal time step according to Theorem 3, ${\delta }_{\exp }$: maximal time step in the numerical experiments.

	$\mathit{\alpha }=0.5$			$\mathit{\alpha }=0.7$			$\mathit{\alpha }=0.9$
h	0.0785	0.0392	0.0314	0.0785	0.0392	0.0314	0.0785	0.0392
${\delta }_{\exp }$	0.0550	0.0270	0.0217	0.0128	0.0045	0.0031	0.0027	0.00083
${\delta }_{\mathrm{theory}}$	0.0555	0.0277	0.0222	0.0132	0.0050	0.0037	0.0032	0.0009

Table 6. Comparison of experimental and predicted stability results by using bilinear finite elements on a uniform grid in Example 2. The maximal time steps ensuring stability are related for different values of h and fractional powers $\alpha$. ${\delta }_{\mathrm{theory}}$: maximal time step according to Theorem 3, ${\delta }_{\exp }$: maximal time step in the numerical experiments.

	$\mathit{\alpha }=0.5$			$\mathit{\alpha }=0.7$			$\mathit{\alpha }=0.9$
h	0.0785	0.0392	0.0314	0.0785	0.0392	0.0314	0.0785	0.0392
${\delta }_{\exp }$	0.0780	0.0383	0.0303	0.0203	0.0075	0.0053	0.0051	0.0014
${\delta }_{\mathrm{theory}}$	0.0785	0.0392	0.0314	0.0215	0.0081	0.0060	0.0059	0.0017

We also note that in all the experiments related to Examples 1 and 3, the numerical solution is non-negative.

3.4. Efficiency

Note that instead of applying fast matrix power–vector products, a number of modern techniques have been proposed to compute the right-hand side of (1) efficiently. A conventional choice is the fast Fourier spectral method [26], which is based on the identity

$$\mathcal{F}\left[{(-\Delta )}^{\alpha /2}u\right]\left(\xi \right)={\left|\xi \right|}^{\alpha }\mathcal{F}u\left(\xi \right),0<\alpha \le 2$$

with $\mathcal{F}$, the conventional notation for the Fourier transform. In the discrete setting, we pursue a similar strategy. The algorithm presented in Section 3.2.1 is based on a suitable truncation of the associated spectral representation, combined with a fast algorithm for approximating the remaining terms. A related approach is the fast convolution method [27], which likewise relies on the discrete Fourier transform.

An alternative class of methods is provided by hierarchical matrix techniques, which are based on low-rank approximations of the off-diagonal blocks of the dense matrix representing the fractional operator [28].

In all approaches mentioned above, the objective is to achieve a computational complexity of $O(N_{x}log{N}_x)$ for a single matrix–vector product, where $N_x$ denotes the number of unknowns, corresponding to the number of interior grid points.

We compare this target complexity with that of our algorithm, as reported in Table 7. Here, $N_t$ denotes the number of time steps, and $n_h$ represents the number of unknowns per spatial dimension. Accordingly, $N_x=n_h$ in Example 1, while $N_x=n_h^2$ in Examples 2 and 3.

Table 7. Computational times (in seconds) in the case of the numerical experiments in Example 1 (one-dimensional case) and Examples 2 and 3 (two-dimensional case). In Examples 1 and 2, a finite difference approximation was used with the first $K=1000$ terms in the Taylor expansion, and we set $k_1=k_2=20$. In Example 3, finite element approximation was used with the first $K=1500$ terms in the Taylor expansion, and we set $k_1=k_2=25$.

Example 1				Example 2				Example 3
${\mathit{N}}_{\mathit{t}}$	${\mathit{n}}_{\mathit{h}}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{6}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{8}$	${\mathit{N}}_{\mathit{t}}$	${\mathit{n}}_{\mathit{h}}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{6}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{8}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{6}$	$\mathit{\alpha }=\mathbf{0}.\mathbf{8}$
1000	12	0.538	0.561	5	6	0.013	0.008	0.018	0.022
1000	24	0.689	0.629	20	12	0.071	0.054	0.120	0.141
1000	48	0.705	0.639	80	24	0.313	0.310	0.742	0.761
1000	96	0.777	0.803	320	48	3.508	3.541	8.737	8.939
1000	192	1.101	1.233	1280	96	64.343	65.27	123.641	125.227

Therefore, in the case of Example 1, linear complexity would imply an increase in the computing time by a factor of two. Similarly, for Examples 2 and 3, linear complexity corresponds to a factor of $4·2^2=16$ between the consecutive computing times. In this light, the observed results would even suggest a sublinear growth of the computational time with respect to the number of unknowns. This conclusion, however, is overly optimistic. The computation of the eigenvectors itself also contributes non-negligibly to the total runtime, and in MATLAB R2024a—even in the optimal, well-preconditioned case—one can at best expect linear complexity. Moreover, the execution of long or nested loops may account for a substantial portion of the overall computational cost, particularly in cases where the total computing time is small.

4. Conclusions and Future Research

In the framework of the matrix transformation method, the stability and the preservation of non-negativity for space-fractional diffusion problems can be analyzed independently from the spatial dimension. The stability condition depends only on the conventional finite difference discretization matrix and the fractional power of the Laplacian. Taking a spatial refinement, this leads to milder stability conditions compared to the case of conventional diffusion. This was confirmed in a series of numerical experiments where one can observe the sharpness of the stability bounds and the robustness of them regarding the spatial dimension. Also, the preservation of non-negativity can be established. In the case of the finite element discretization, the key step for an efficient algorithm is the application of mass lumping. In this way, the related simulations can be performed only using sparse matrix–vector products. From a practical standpoint, the efficient implementation of implicit time-stepping schemes and the extension of the streamline diffusion finite element method to fractional Laplacian operators constitute worthwhile directions for future research.