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

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The first conceptual flaw in the current version of the authors' work, which will catch the eye of any potential reader, is the complete inconsistency of the title of the work and its abstract with the real content of the work. From the title of the paper and its abstract, any potential reader will conclude that the authors' work is devoted to an important applied problem - the development of numerical methods for solving partial differential equations (which include the diffusion problems), which would have an important practical property (preserving the positivity of an approximate numerical solution in the event that an analytical solution must be positive). However, later in the introduction and at the beginning of the main part of the work (Section 2), it turns out that the authors' work is not devoted to the development of a general numerical scheme for solving problems in this case, but only for a specific model example. Any potential reader will reasonably have the feeling that the authors have misled him / her. Can the authors' method be generalized to solve a whole class of problems, which is included in the title of the work and in its abstract? If this question had a positive answer, then most likely the authors would immediately consider and justify a general approach to an approximate numerical solution of this class of problems. Therefore, the title of the work and its abstract should be clarified so that they correspond to the actual content of the work. But in this case, another problem arises - the authors' work will give the impression of a textbook example, rather than a full-fledged research article. This will be due to the fact that the paper describes only approaches to solving a specific synthetic problem (1), which is unlikely to be in demand when solving real-world applied problems (at least, the authors do not provide references justifying the relevance of the problem under consideration).

From a mathematical point of view, the article is written rather carelessly - there is a feeling that the authors did not reread it before submitting it and/or did not work out the research material sufficiently. Some examples of such mathematical carelessness:

1. At the very beginning, there is an obvious typo in the first equation of problem (1) - a sequence of multiplication and subtraction operations. Which of these operations is listed in vain? What about subtraction?

2. In the introduction, the authors refer to the implicit Euler and Crank-Nicholson schemes, but do not mention the Rosenbrock one-stage scheme with a comemplex coefficient (CROS1), which has significant advantages over these schemes: $L_2$-stability, second-order accuracy, monotony (positivity preserving). Thus, the introduction is not complete and does not reflect the current state of affairs in the subject under consideration. Moreover, a potential reader may think that the authors are using outdated research methodology. Therefore, the fact that the methodology in question does not use the CROS1 scheme should be clearly motivated.

3. The statement ``For the simplicity, we will set $\mu = 1$ in the upcoming analysis'' is doubtful, since for sufficiently small values of diffusion coefficient, the solution to the problem under consideration may have significant features - it may contain internal and/or boundary layers.

4. The remark that ``$A_{FD}$ is positive definite, which is satisfied in the case of classical finite differences'' will not be obvious to a potential reader and requires careful mathematical justification (for example, it is obvious that in the case of an upward convex function, the corresponding value will be negative). Perhaps the authors forgot to formulate any other conditions under which the corresponding statement would be true?

Disadvantages of this kind are found throughout the article (the above are only examples of such disadvantages).

Thus, the article requires significant revision.

Author Response

First of all, we thank you for the careful reading of the manuscript and for the valuable suggestions.

Comment A1:

Reply:

In this work, we did not want to develop new time steppings in the numerical schemes. Instead, we raised the question which of the conventional ones preserve positivity if they applied to space fractional diffusion problems. Also, we investigated the question, how the stability properties of the classical schemes are changing if they applied to the numerical solution of space fractional diffusion problems instead of the conventional diffusion problem. Changing the fractional power, can we obtain then less strict stability constraints?

We admit that in this sense the title and some parts of the introduction can be misleading, but in any case, we did not want to mislead anyone. To improve the clarity of the work from this point of view, we have revised and extended now the explanation.

In concrete terms:

We have changed the title.
We have inserted a new second sentence in the abstract to clarify the above issue.
Also, in lines 51-52, in the introduction, we have stressed this issue.

Comment A2:

But in this case, another problem arises - the authors' work will give the impression of a textbook example, rather than a full-fledged research article. This will be due to the fact that the paper describes only approaches to solving a specific synthetic problem (1), which is unlikely to be in demand when solving real-world applied problems (at least, the authors do not provide references justifying the relevance of the problem under consideration).

Reply:

Yes, we agree: real-world problems are more complex, often including additional terms—advection, reaction, etc.—with coefficients that can be extremely large or small. Incorporating these terms, provided we have a well-established treatment of the fractional Laplacian, is relatively standard. However, the efficient treatment of the fractional Laplacian itself is far from beeing straightforward. Most authors still rely on full matrices, which, in practical applications, lead to prohibitively high computational costs. Therefore, we focused on the most challenging component of such problems—the fractional operator—and aimed to develop a clear and efficient solution algorithm, accompanied by a rigorous mathematical analysis. Also, as pointed out in the new section 3.4, the present simple approach is as efficient as the recent (and more involved) approaches in the literature. (Incorporating this comparison was suggested by the other reviewer.)

Nevertheless, this an an important point such that we have inserted a new sentence in lines 85-87.

Comment 1:

At the very beginning, there is an obvious typo in the first equation of problem (1) - a sequence of multiplication and subtraction operations. Which of these operations is listed in vain? What about subtraction?

Reply:

This is not a typo, we explain it hereby in details, how to understand it.

Take -Δ, which with Dirichlet boundary conditions, is a positive self-adjoint operator with a compact inverse.
Take (-Δ)^α, which makes sense for any non-negative α, but we use the case α∈(0,1] in the analysis.
Take -(-Δ)^α, which for α=1, gives the conventional Laplace operator.
Finally, using the diffusion coefficient µ, we obtain µ· -(-Δ)^α as given in (1).

Nevertheless, we have revised this formalism and moved the minus sign to the beginning to improve readability.

See the new version of (1).

Comment 2:

In the introduction, the authors refer to the implicit Euler and Crank-Nicholson schemes, but do not mention the Rosenbrock one-stage scheme with a complex coefficient (CROS1), which has significant advantages over these schemes: $L_2$-stability, second-order accuracy, monotony (positivity preserving). Thus, the introduction is not complete and does not reflect the current state of affairs in the subject under consideration. Moreover, a potential reader may think that the authors are using outdated research methodology. Therefore, the fact that the methodology in question does not use the CROS1 scheme should be clearly motivated.

Reply:

It is indeed true that implicit time-stepping schemes, such as CROS1—a generalization of the implicit Euler method—exhibit optimal stability properties. Still, motivated by practical simulations, we have placed the emphasis on explicit methods. The reason is as follows. Applying an implicit time step in the presence of the fractional Laplacian operator requires solving a linear problem involving a matrix power A^α. At the same time, we aim to avoid computing even this matrix, as it is dense. Notably, the technique described in Section 3.2.1 makes this avoidance possible in the case of explicit approaches. Finally, to complete the discussion, we now mention alternative approaches, such as CROS1, which can improve stability and conservation properties.

See lines 28-30 and 182-184 in the revised version.

Remarks:
1. We also have ideas on how to avoid numerically solving a linear system in which the system matrix is not given explicitly. GMRES-based approaches, combined with the technique described in Section 3.2.1, appear to be a promising candidate for this purpose.

2. I am also working in room acoustics: we simulate with 10-20 millions of unknowns and we need to perform at least 10 thousands time steps. At the present computing capacities, this needs explicit time steppings.

Comment 3:

``For the simplicity, we will set $\mu = 1$ in the upcoming analysis'' is doubtful, since for sufficiently small values of diffusion coefficient, the solution to the problem under consideration may have significant features - it may contain internal and/or boundary layers.

Reply:

We completely agree. Accordingly, we have added an additional remark in the revised manuscript. In this setting, one would most likely need to employ an extension of the classical streamline-diffusion finite element method (SDFEM) to problems involving the fractional Laplacian. To the best of our knowledge, we are not aware of any fully satisfactory references addressing this issue; nevertheless, this appears to be a very interesting and worthwhile direction for future research.

See lines 85-87 and 457-460 in the revised text.

Comment 4:

The remark that ``$A_{FD}$ is positive definite, which is satisfied in the case of classical finite differences'' will not be obvious to a potential reader and requires careful mathematical justification (for example, it is obvious that in the case of an upward convex function, the corresponding value will be negative). Perhaps the authors forgot to formulate any other conditions under which the corresponding statement would be true?

Reply:

We still state that any reasonable finite difference approximation of the operator -Δ (with homogeneous Dirichlet boundary conditions) results in a positive definite matrix A_FD.

We stress here, that this is a property of the corresponding matrix. Also in that case, Av can have negative components even if v is positive componentwise (we use Matlab convention):

[2 -1 0; -1 2 -1;0 -1 2] * [1 3 1]' = [-1 4 -1].

Here the matrix is positive definite, but we have applied a vector, which can be identfied as a discretization of a concave function.

If we were to include a paragraph/proof on this issue, one could argue that we are merely restating well-known facts readily available in classical textbooks.

At the same time, it is really important to state this issue clearly, supported with a mathematical justification. Therefore, we have rather included a classical reference.

See the revised text in lines 169-170 and the corresponding reference [23].

Reviewer 2 Report

Comments and Suggestions for Authors

Kindly, see the attached report

Comments for author File: Comments.pdf

Comments on the Quality of English Language

English requires moderate editing (sentence structure, clarity).

Author Response

First of all, we thank you for the careful reading of the manuscript and for the valuable suggestions.

Comment 1: The paper does not clearly articulate what new theoretical contribution or computational advantage it provides beyond existing studies, and the comparison with current numerical methods in the literature remains insufficent.

Response: In the Results section, all statements (Theorems 2–5) are, to the best of our knowledge, new. The main departure from classical studies lies in our investigation of the fractional Laplacian operator.

The principal difficulty in this setting is that the corresponding matrix power is not computed explicitly; nevertheless, it is necessary to establish estimates for its stability properties and its ability to preserve positivity. We admit that the tools we have used are really classical. Note also that we had a careful search whether someone did use the key reference [20] for stability analysis or in general, for the analysis of space-fractional diffusion: we did not find any relevant study. Moreover, as we stressed in the introduction, some previous studies have limited results: either the dependence of the fractional power or the space dimension.

To point out this fact even more clearly, we have rewritten the introduction; see lines 59-66 in the new version.

Also, the computational algorithm, which combines the matrix power-vector product (without computing the matrix power itself) with the mass lumping technique is new and paves the way for efficient finite element-based simulations.

Comment 2: The paper would benefit from a concluding discussion comparing efficiency against other modern fractional solvers (e.g., fast convolution quadrature, spectral methods).

Reply: As proposed, we have performed new numerical experiments (see the forthcoming comment) and the experimental results were discussed.

See the completely new section 3.4.

Comment 3: Error analysis does not include computational efficiency or complexity comparisons, which are important for “Algorithms”.

Reply: This is strongly related with the previous point and was really missing. To fill this gap, we have created section 3.4 with numerical experiments and comparison.

Comment 4: English requires moderate editing (sentence structure, clarity).

Reply: We have carried out a thorough revision of the manuscript with respect to the English language. The principal changes are highlighted in the revised version.

Comment 5: The introduction is lengthy and repeats previously known facts without synthesizing the main contribution.

Reply: Following the suggestion, we retained certain parts of the Introduction and merged others. In particular, the contribution of the present work is now presented in the last paragraph. Several new points were incorporated based on the comments of the other reviewer.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Below I will repeat the comments I made about the previous version of the article, and after each comment I will indicate in bold my opinion about the work done by the authors to improve the material of the work.

The first conceptual flaw in the current version of the authors' work, which will catch the eye of any potential reader, is the complete inconsistency of the title of the work and its abstract with the real content of the work. From the title of the paper and its abstract, any potential reader will conclude that the authors' work is devoted to an important applied problem - the development of numerical methods for solving partial differential equations (which include the diffuson problems), which would have an important practical property (preserving the positivity of an approximate numerical solution in the event that an analytical solution must be positive). However, later in the introduction and at the beginning of the main part of the work (Section 2), it turns out that the authors' work is not devoted to the development of a general numerical scheme for solving problems in this case, but only for a specific model example. Any potential reader will reasonably have the feeling that the authors have misled him / her. Can the authors' method be generalized to solve a whole class of problems, which is included in the title of the work and in its abstract? If this question had a positive answer, then most likely the authors would immediately consider and justify a general approach to an approximate numerical solution of this class of problems. Therefore, the title of the work and its abstract should be clarified so that they correspond to the actual content of the work.

In the new version of the article, the authors have added the necessary changes, as a result of which potential readers will not be misled.

In the new version of the article, the authors have justified the motivation for choosing the main task for consideration. Now the work does not give the impression of work, the choice of the main task in which is motivated only by the convenience of applying theoretical methods.

The authors explained that there was no typo in the original version of the article. The typo suggested by the reviewer was a feature of the notation. However, the authors have brought the designations to the generally accepted ones in order not to confuse arbitrary readers from a fairly wide range who are not familiar with the relevant specific designations.

The authors of the article motivated the choice of a numerical scheme.

In the new version of the article, the authors have added comments regarding this fact.

Thus, my opinion is that the manuscript has been sufficiently improved to warrant publication in Algorithms.

Reviewer 2 Report

Comments and Suggestions for Authors

Now, the article slightly improved