I stopped working in this field nearly 20 years ago. But even than there existed several methods that outperformed RK4 by lengths, the comparison is done on the estimated accuracy obtained by the same number of calls to the ODE's right hand side function. Have a look in the following books

Hairer, E. and Nørsett, S. P. and Wanner, G.: Solving Ordinary Differential Equations, Part I - Nonstiff Problems, Springer Verlag, 1987.

Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations, Part II - Stiff and Differential-Algebraic Problems, Springer Verlag, 1991.

There the methods (for non-stiff ODE) of Dormand/Prince are highly recommended (I used one in my time). Somewhere in the middle of the first book is a twosided graphics where the orbits of a specific problem are shown obtained by different methods (and the same maximum number of rhs calls). The orbits of the chosen ODE are known from theory to return to the starting point. The impressive fact of the graphics is to show how good this feature is reproduced by the methods applied: the orbit of 1-step Euler metod leaves the pages and does not return, the orbit of the RK4 method leaves a large gap between start and end points, the orbit of the DP method without stepsize control leaves a small gap, and the orbit of DP with stepsize control leaves no visible gap; even more, the last method needed much less rhs calls!

This was state of the art 20 years ago. Maybe that meanwhile better methods have been developed, maybe the initially discussed PAM is one. Anyway, I recommend that you replace the RK4 method in your code at least by the DP4/5 method. If you use Mathlab then apply ode45 as ODE solver, if your code is in Fortran or C then search the internet for DOPRI5.

Preprint Piecewise Analytic Method (PAM) is a New Step in the Evoluti...

PAM method.

PAM method is more accuracy than Runge Kutta method.

How accurate is PAM? Does it have a formal order of accuracy as with Runge Kutta methods?

How easy is it to implement for a highly nonlinear set of ODEs? One of the reasons I ask this is that a recent paper of mine solved 3605 first order ODEs using the 4th order RK scheme and the shooting method. This was a boundary value problem with 3591 unknown initial conditions which had to be found using the Newton-Raphson scheme which is embedded in all shooting method codes. Would PAM be able to solve this, and if so, how long would it take to write a code to do so?

Dear Prof. D. Andrew S. Rees

I’ll answer your questions one by one:

1st question: How accurate is PAM?

Answer: The accuracy of PAM is unlimited you can control in the accuracy according to your needs and capabilities of your devices.

You can see Preprint Piecewise Analytic Method (PAM) is a New Step in the Evoluti...

Many thanks, sir. The same is true for Runge-Kutta methods. One may decrease the step length and increase the order of accuracy. I am thinking specifically of how quickly does the solution converge as a function of step lengths.

Regards,

Andrew Rees

D. Andrew S. Rees

2nd question: Does it have a formal order of accuracy as with Runge Kutta methods?

Answer: Yes, it has. In Runge Kutta you adapt the parameters to match with Taylor series according to the order of accuracy you need but in PAM you don’t need to adapt any parameters you calculate the terms directly according to the orders you need. Also, in PAM you can obtain accuracy which you can not obtain by RK.

You can see Preprint Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study)

D. Andrew S. Rees

3rd question: How easy is it to implement for a highly nonlinear set of ODEs?

Answer: It is very easy and accurate please see the paper

Preprint Solving Nonlinear 2 nd Order Differential Equations Using Pi...

It is used to solve highly nonlinear DE.

Also, if you have another highly nonlinear DE you can send it to me and test my solution and you are welcome to criticism the solution if you have better solution. I'm here to learn fro you and the others not to win.

D. Andrew S. Rees

4th question: One of the reasons I ask this is that a recent paper of mine solved 3605 first order ODEs using the 4th order RK scheme and the shooting method. This was a boundary value problem with 3591 unknown initial conditions which had to be found using the Newton-Raphson scheme which is embedded in all shooting method codes. Would PAM be able to solve this, and if so, how long would it take to write a code to do so?

Answer: If your problem is solved using RK, I’m sure that PAM will solve your problem and gives you more accurate results. How long it takes I don’t know because I didn’t see your problem.

D. Andrew S. Rees

Thank you for your questions. I’m here to learn from you and the others.

Please if you have any negative feedback or criticism, don’t hesitate to write it.

Need your supports, comments and announcement for my method.

Best regards,

Tamer Abassy

Many thanks again. I agree with your last point - RK is not my method and doesn't belong to me; I have merely used it for a long time and have streamlined my approach to solving ODEs with it. If there is something else out there that is superior in some way, then I always investigate. There are also methods other than RK for ODEs of different types - one example is the Dormand-Prince family which are used for stiff equations. Others, for boundary value problems, are direct methods and can be solved using ordinary 2nd order finite differences, or even compact differences, and if the problems are nonlinear then a Newton-Raphson scheme can be applied. Periodic solutions too. Spectral methods. Many things, including problems with multiple eigenvalues.

But I have one further question. Is PAM a method for which a black-box solver may be written? My own RK boundary value problem solver (which admittedly has been developed gradually over some decades) has one subroutine to define the system of first order equations, one to define the conditions which need to be satisfied, another which declares which initial conditions are unknown and another which declares the values of all the initial conditions (both known and guesses for the unknowns). As it is a boundary value problem a Jacobian matrix needs to be defined for the multi-dimensional Newton-Raphson scheme - this is computed numerically within the code without a need for case-specific programming. So this coding structure means that a new problem may be programmed very quickly indeed, a matter of minutes sometimes, and hence a first solution (prior to checking for accuracy, of course) can sometimes be very quick. So if someone were to give a PAM used a new system of ODEs to solve, perhaps it will be a 10th order nonlinear system of first order equations with, say, eight known initial conditions and two unknown, where the two remaining conditions are nonlinear (e.g. variable1*variable2=1 at t=tmax) then is it possible for a black-box form of PAM to exist which can be programmed in a matter of minutes to solve this?

One reason for asking this question is that, should one require 10 such computations for a research paper, then my preference would be to use a quick familiar scheme. But if I were to require 10,000 such computations, then I would be prepared to invest time to program something superior if there is something superior to program.

I certainly don't have negative feedback for you, but I always worry when I hear strong claims! The concept of one method being more accurate than another is ill-defined until a constraint is given. Perhaps that constraint might be CPU time - for a given CPU time which method gives more significant figures? Between PAM and RK I do not know the answer. Without sitting down and doing a few calculations, I don't know whether RK2 is better than RK4 for a fixed number of FLOPS. Because the number of FLOPS per timestep is fewer for RK2 than it is for RK4, smaller timesteps can be used for RK2 than for RK4, but will RK2 be more accurate. I would guess RK4, but these are important points for everyone to think about.

I am guessing that those who follow me on RG will have been informed about my contribution to this discussion, so this is publicity for PAM.

Many methods can compete and overwhelm Runge-Kutta algorithms.

Can you provide a working codes e.g. for Rossler system? I can experimentally check the performance and convergence of your method on different data types.

Very interesting!

I looked at your paper but could not understand the scheme at the first glance.

It would be good if you could provide a Matlab or Python script for a simple single DOF example. It will also help assess the computational cost for those interested.

Dear Prof. D. Andrew S. Rees ,

Your question: Is PAM a method for which a black-box solver may be written?

Answer: I’m a programmer and I have my own black-box. I can’t publish it now. If any software company wants to buy it’s OK.

Now if anyone needs a quick solution and save time, I can cooperate with him and work together. I work on my code since 10 years.

D. Andrew S. Rees

You have all the rights to say“ I always worry when I hear strong claims!.....”

It isn't a good behavior what I did. I must leave the judgment to the audience but I have my reasons to do this.

I tried many times to publish my work in respected journals but the papers returned back to me without referee...

I'm exhausted; I doubted in my work, I have no groups and work alone. Also I'm not ready to write names of other persons on my work to publish my work in respected journals. Also, I have found others took my ideas and publish it and didn't refer to me in spite of they are follow my work.

I have achieved an additional work with excellent results. So I decided to burn some of my work and publish it on Research gate and get audience.

I think I have succeeded up to now and I didn’t have any negative feedback or criticism.

Best regards,

Tamer Abassy

Dear Denis Butusov ,

You said “Many methods can compete and overwhelm Runge-Kutta algorithms.”

When I said is there a method can compete RK?

I specified a name of the method (PAM) and enhance what I said through results and papers. I opened the door to criticize the method and learn from all of you.

Please, specify name and scientific result.

Denis Butusov

Your question: Can you provide a working codes e.g. for Rossler system? I can experimentally check the performance and convergence of your method on different data types.

Answer: You before suggest to test my method and I said you are welcome to do this and you do nothing.

You repeat what you said here again but now you demand my code which I have worked on it since 10 years.

Instead of saying give me the code you must say we can cooperate in solving Rossler system and publish the results if it is good.

I attached photo from your previous offer and Photo of Quick result of Rossler system using PAM.

Dear Chennakesava Kadapa

You did not understand the scheme at the first glance. Do you thing that you will understand my method from the code.

If you or anyone needs a quick solution and save time, I can cooperate with him and work together. I work on my code since 10 years.

regards,

Tamer Abassy

I appreciate that you have worked on this for 10 years. If you want to popularise your method, then you have to provide the code, at least for a sample problem. As you know, we are all busy with own work. Not everyone would like to go through the paper for days and then implement the scheme themselves.

Besides, many a times it is difficult to understand the scheme from the paper because of the use of different notations by different researchers. If someone wants to understand your method, it would help them if you can provide the code for a simple sample problem. By doing so, you don't have to work with everyone who is interested in learning your method. They can simply run the code for the example problem and try to understand the steps involved without much difficulty.

You can do something similar to what I did here

https://ckadapa.wordpress.com/2017/07/20/numerical-solutions-of-daes/

Dear, Chennakesava K.

I agree with your answer.. good luck.

I stopped working in this field nearly 20 years ago. But even than there existed several methods that outperformed RK4 by lengths, the comparison is done on the estimated accuracy obtained by the same number of calls to the ODE's right hand side function. Have a look in the following books

Hairer, E. and Nørsett, S. P. and Wanner, G.: Solving Ordinary Differential Equations, Part I - Nonstiff Problems, Springer Verlag, 1987.

Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations, Part II - Stiff and Differential-Algebraic Problems, Springer Verlag, 1991.

There the methods (for non-stiff ODE) of Dormand/Prince are highly recommended (I used one in my time). Somewhere in the middle of the first book is a twosided graphics where the orbits of a specific problem are shown obtained by different methods (and the same maximum number of rhs calls). The orbits of the chosen ODE are known from theory to return to the starting point. The impressive fact of the graphics is to show how good this feature is reproduced by the methods applied: the orbit of 1-step Euler metod leaves the pages and does not return, the orbit of the RK4 method leaves a large gap between start and end points, the orbit of the DP method without stepsize control leaves a small gap, and the orbit of DP with stepsize control leaves no visible gap; even more, the last method needed much less rhs calls!

This was state of the art 20 years ago. Maybe that meanwhile better methods have been developed, maybe the initially discussed PAM is one. Anyway, I recommend that you replace the RK4 method in your code at least by the DP4/5 method. If you use Mathlab then apply ode45 as ODE solver, if your code is in Fortran or C then search the internet for DOPRI5.

Dear Saad Najeeb , Chennakesava Kadapa and all of you

I try to search for a software company to give them my code and experience for introducing a tool like ode45 .

If you have any suggestion and experience in that field please help me.

Up to this happen, you can write your code or communicate with me to give you a solution and save your time and effort.

Best regards,

Tamer Abassy

you can compare RK4 with any method of order 4, RK4 method can be compared with any other method of order 4 . But always pay attention to the cost.

Dear Muhannad Shallal ,

I said compete not compare.

Regards,

Tamer Abassy

Hi Tamer,

I am not sure what you mean about competing and comparing, and therefore I don't understand the purpose of this thread!

In terms of running, I can compete with Sir Mo Farah, but the comparison is that he is very much faster. The former is trivial but yields no information, while the latter is informative.

With that analogy in mind, shouldn't "compare" be the better test?

Andrew

Hi Prof. D. Andrew S. Rees

English is not my native language ….. So, you may be wright.

What I mean by Compare and Compete is :

Compare, you can compare between any two things even if they are opposite like strong and weak.

But when you compete, you must compete in the same level or better to win or to improve your performance and feel excitation. (like Real Madrid, Bacelona, Liverpool and Manchester United football teams).

Thank you,

Best regards,

Tamer Abassy

I reckon that we are both right, then! Much depends on the analogy used.

Dear Dr. Tamer

Sorry . (predictor-corrector: Adams-Bashforth-Moulton ) is one of multistep method can be a good competitor to RK4.

regards

Looking back at a ODE program that I used several decades ago, my software notes / comments indicate that Runge-Kutta combined with predictor-corrector approximation yields better / more accurate results, but takes much longer to run.

I agree with your answer Mr. Muhanned Shallal.

Dear Muhannad Shallal , Saad Najeeb and Arnold Liu ,

It is good to read about methods can compete RK.

If you are expert in these methods, we can cooperate and write another comparative study between PAM and your methods.

Regards

Tamer Abassy

Dear All, I wrote in the previous answers that" I suffer that my papers about PAM are returned back without review".

Finally, a journal returned my paper with reviewer comments.

Please, see the attached paper file and the attached journal reply file.

In the following answers, I'll reply on the reviewer comments and proof scientifically that the reviewer is unfair and didn’t read the paper.

My reply will be written using bold font

The reviewer said" The manuscript at hand is entitled "Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study)" and presented by T.A. Abassy. In this paper, the author pretends to presenting a numerical integration tool that outperforms the existing ones. Certainly, this is untrue. From my point of view, the research under consideration is too weak and of no theoretical and/or practical interest because of the following reasons:"

My reply:

I think the reviewer must say “pretend that” not “pretend to”.

I didn’t talk about numerical integration at all in the paper. I think the reviewer was reviewing another paper about numerical integration when he reviewed my paper or the reviewer doesn’t know the difference between integration and differentiation.

The paper is a comparative study between PAM and RK not between PAM and existing ones.

When the reviewer said "untrue" I think the reviewer must mention the wrong result, but unfortunately he didn’t mention my mistakes to learn from him.

I respect the reviewer point of view according to his interests.

The reviewer said that my research is too weak, so I’ll ask him to give me an existing numerical method which he can use to solve case study 2 in my research and compare his result with my results.

The reviewer said " 1. The novelty of the manuscript at hand is questionable because the Piecewise Analytic Method (PAM), which is highlighted as the main contribution of the author, has been already published in [5-9]. Thus, this novelty is limited only to some comparison to a number of explicit Runge-Kutta methods, which is extremely poor and, certainly, insufficient for approving its publication in the journal of computational and applied mathematics."

My reply:

The [5-9] papers are my papers and all of them are introduction to my method PAM. They are not a comparative study like the reviewed paper and the reader can judge.

The reviewer said "my examples are extremely poor". My reply “If they are poor and the comparative study in my paper shows that my method PAM is powerful than RK which is famous powerful method. I think it is a sufficient reason to publish it to announce that there is another new method is better than the most famous method RK…

I ask the reviewer to give me his powerful example to solve it by RK and compare the results with PAM results.

The reviewer said " 2. The PAM is unclear and poorly presented in the paper under review. This method is grounded in the spline interpolation idea and outline briefly in Section 3. Unfortunately, the author's presentation is somewhat obscure for readers' understanding how the piecewise polynomial (10) is constructed and applied.

My reply:

PAM is clear and good presented in [5-9].

Does the reviewer needs me to repeat [5-9] in this paper to understand my method? In spite of he said in comment 1 this paper is a repeat of [5-9].

This reviewer point 2 emphases that he didn’t understand my method. In spite of this he said my results are wrong and my work is previously published without proof.

The reviewer said " 3. The author claims that the PAM "is a middle method between numerical and analytic methods and solves critical problems in both". Certainly, this is untrue. This technique implies a huge amount of users' hand work, which makes this approach highly ineffective for solving stiff and large-scale problems. Note that the numerical tests presented in Section 4 cover the trivial ODE of size one, only. I would definitely recommend the author to compare the RAM with advanced Runge-Kutta solvers on the set of challenging test problems collected in [E. Hairer, G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, 1996, p. 144-152], for instance.

My reply:

Dear reviewer, Please, read my paper

https://www.researchgate.net/publication/332910969_Piecewise_Analytic_Method_PAM_is_a_New_Step_in_the_Evolution_of_Solving_Nonlinear_Differential_Equation

to know if my method is "a middle method between numerical and analytic methods and solves critical problems in both".

I think the journal chose a wrong reviewer because numerical or computational mathematics based on computer software not hand work.

The reviewer didn’t understand my method with simple problems as he wants me to use my method with more complicated problem. Will the reviewer understand my method with more complicated problem? Again, the PAM method is new and it is logic to go step by step and the difficulty is increasing gradually.

Does the reviewer want me to write a book and solve very complicated problem directly?

I want to thank the reviewer for opining new point which is comparing my method with advanced Runge-Kutta solvers which I’ll do in my next steps, but when I'll do this, I’m sure that I’ll obtain an excellent results using PAM.

The reviewer said " 4. Finally, the paper of T.A. Abassy focuses on only the fixed-stepsize numerical integration, which is hardly effective. Such numerical integration tools are almost useless in practical computations and not competitive to good adaptive ODE solvers with automatic local and global error control mechanisms. I am absolutely sure that the constructed PAM loses even the standard Matlab codes in the accuracy and efficiency of solving various ODEs, and these Matlab ODE solvers are commonly available. "

My reply:

Again, I didn’t write a book to cover all the points and the PAM method is new so I must start step by step to spread the method and the readers understand it.

Dear reviewer, Again, I didn’t use integration please you must know the difference between differentiation and integration and also know that one of the most important point of the basics of scientific research is working step by step (variable step size is next future work…)

Because I’m a researcher and knows the basics of scientific research. I’ll not talk like you without proof.

Please give me any example you solved with Matlab and I’ll solve using PAM and compare your results with my results. I’m sure PAM will do better.

The reviewer said " In conclusion, the manuscript at hand is poorly written and contributes neither to the theory of numerical methods for ODE nor to the practice of their efficient implementation. Its novelty is also questionable. That is why I strongly object publication of the mentioned manuscript under review in the journal of computational and applied mathematics."

My reply:

In conclusion, no additional comments. I think I replied on this in the previous comments.

Dear All, Please, don't underestimate my work without proof.

Finally, I didn’t say that " I didn’t make mistakes". Of course, I have mistakes. So, please correct my mistakes scientifically and correct my language mistakes also because English is not my native language.

Your comments are welcome.

Best regards,

Tamer Abassy