Denys DUTYKH's
2017-02-28T11:08:04+00:00
http://www.denys-dutykh.com/
Denys Dutykh
Applied and Computational Mathematics for Kazakhstan
2017-02-28T00:00:00+00:00
http://www.denys-dutykh.com//2017/02/28/KZworkshop
<p>From 20th to 25th of February 2017 I participated in a workshop organized by the <a href="http://www.kaznu.kz/en/334/page/%20Departments/Faculty_of_Mechanics_and_Mathematics_">Faculty of Mechanics and Mathematics</a> of the <a href="http://www.kaznu.kz/en/">Al-Farabi Kazakh National University</a>. The goal of this workshop was mainly to discuss a new Bachelor degree Programme <em>Applied and Computational Mathematics</em>. The particularity of this Programme is that all courses are going to be taught in <strong>English</strong>. This educational programme is a part of the “<strong>Kazakhstan 2050</strong>” Strategy.</p>
<p>Among the participants (besides the <em>Author</em> of this post) we had the pleasure to count the following experts:</p>
<ul>
<li><a href="https://www.researchgate.net/profile/Sergey_Cherny">Sergey Cherny</a>, Institute of Computational Technologies, SB RAS, Russia</li>
<li><a href="http://ait.mtas.ru/en/about/personal/gasnikov.php">Alexander Gasnikov</a>, Moscow Institute of Physics and Technology, Russia</li>
<li><a href="http://www.math.nsc.ru/LBRT/u2/kabanikhin.html">Sergey Kabanikhin</a>, Institute of Computational Mathematics and Mathematical Geophysics, SB RAS, Russia</li>
<li><a href="https://www.keele.ac.uk/scm/staff/professors/juliuskaplunov/">Julius Kaplunov</a>, Keele University, UK</li>
<li><a href="https://www.researchgate.net/profile/Matthias_Meinke">Matthias Meinke</a>, RWTH Aachen, Germany</li>
<li><a href="http://research.me.udel.edu/~lwang/">Lian-Ping Wang</a>, University of Delaware</li>
</ul>
<p>The conclusions and recommendations issued during this workshop are going to be published in workshop materials to make outcomes available for other Kazakh (and <em>not only</em> Kazakh) Universities.</p>
<p><img src="/public/pics/KZ_crop.jpg" alt="KazNU Workshop" /></p>
<hr />
Fast and accurate computation of cnoidal waves
2017-02-15T00:00:00+00:00
http://www.denys-dutykh.com//2017/02/15/FastAccurate
<p>There are many points to mention and describe briefly today.</p>
<p>First of all, together with my collaborator <a href="http://math.unice.fr/~didierc/">Didier Clamond</a>, we submitted a manuscript, which describes a new method for the computation of cnoidal waves in the full Euler equations with free surface:</p>
<ul>
<li>D. Clamond & <strong>D. Dutykh</strong> <a href="https://hal.archives-ouvertes.fr/hal-01465813/">Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth.</a> Submitted, 2017</li>
</ul>
<p>These solutions can be used to validate and check the accuracy of various dynamic solvers for the full water wave problem. Moreover, the Matlab code is freely available at this URL:</p>
<ul>
<li><a href="https://github.com/dutykh/SSGW/">https://github.com/dutykh/SSGW</a></li>
</ul>
<p>This Matlab code computes irrotational 2D periodic steady surface pure gravity waves of arbitrary length in arbitrary depth. The formulation is based on the so-called Babenko equation and pseudo-spectral discretization in the conformal domain. The resulting equation is solved using Petviashvili iteration method.</p>
<p>In the continuation of our previous short <a href="https://hal.archives-ouvertes.fr/hal-01388481/">publication</a> on peaked solitary capillary-gravity waves, we submitted the full manuscript, which investigates this system of equations (capillary-gravity Serre model) in more details:</p>
<ul>
<li><strong>D. Dutykh</strong>, M. Hoefer, D. Mitsotakis. <a href="https://hal.archives-ouvertes.fr/hal-01465356/">Solitary wave solutions and their interactions for fully nonlinear water waves with surface tension in the generalized Serre equations</a>. Submitted, 2017</li>
</ul>
<p>In the last manuscript we focus more on the dynamics and properties of smooth solitary wave solutions in various regimes with respect to the Bond number.</p>
<p>Finally, the following week I am going to spend at <a href="http://www.kaznu.kz/en/">Al-Farabi Kazakh National University</a></p>
<p><img src="/public/pics/AlFarabi.jpg" alt="KazNU" /></p>
<hr />
Open source problems
2017-01-28T00:00:00+00:00
http://www.denys-dutykh.com//2017/01/28/OpenSource
<p>I would like to share with you a pretty interesting talk given by <a href="http://wstein.org/">William Stein</a>, who is the creator and the main developer in the <a href="http://www.sagemath.org/">SageMath</a> open source project. You can watch the video here:</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/6eIoYMB_0Xc" frameborder="0" allowfullscreen=""></iframe>
<p>The slides can be downloaded separately <a href="http://wstein.org/talks/2016-06-sage-bp/bp.pdf">here</a>.</p>
<p>Personally, I am not a user of SageMath, however, I find that the speaker rises the right questions about open source development and he points out difficulties that the speaker (and his colleagues) encountered on this way.</p>
<p>Just to give some idea about the content, I give here a couple of quotations from this talk:</p>
<ul>
<li>“Every great open source math library is built on the ashes of someone’s academic career.”</li>
<li>“I can’t figure out how to create Sage in academia. The money isn’t there. The mathematical community doesn’t care enough. The only option left is for me to build a company.”</li>
</ul>
<p>I can only endorse the speaker on these points. The software development effort is clearly undervalued in Academia. The general opinion is that if you are a really good mathematician you build a theory. And implicitly, if you are mediocre, you just code. This inadmissible attitude has to change.</p>
<hr />
Two interesting talks
2017-01-21T00:00:00+00:00
http://www.denys-dutykh.com//2017/01/21/Talks
<p>I would like to share with you a couple of videos coming from the <a href="http://www.engin.umich.edu/college/">Michigan Engineering</a> (<a href="https://www.umich.edu/">University of Michigan</a>) that I find particularly interesting and informative. The first one is given by <a href="https://en.wikipedia.org/wiki/Philip_L._Roe">Phil Roe</a>, a well-known person in the finite volume community. He gives a critical account of the state-of-the-art in CFD, meshes and existing problems. This talk is to listen absolutely if you are interested in computations.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/uaH91P665PI" frameborder="0" allowfullscreen=""></iframe>
<p>As a side remark, there is a pretty interesting report produced by NASA on a very close topic:</p>
<ul>
<li><a href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20140003093.pdf">CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences</a></li>
</ul>
<p>And as promised, the second video is devoted to common misconceptions in aerodynamics. It is delivered by Doug McLean who made his career at Boeing. To listen absolutely, if you want to understand truly, for example, why airplanes fly (but not only):</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/QKCK4lJLQHU" frameborder="0" allowfullscreen=""></iframe>
<p>Have a pleasant listening!</p>
<hr />
Heat and Mass Transfer (HAM)
2017-01-20T00:00:00+00:00
http://www.denys-dutykh.com//2017/01/20/HAM
<p><img src="/public/pics/Heat.jpg" alt="Heat" /></p>
<p>I would like to share here two recent publications which resulted from my collaboration with one Brazilian (<a href="http://www2.pucpr.br/educacao/lst/">LST, PUCPR</a>) and one French (<a href="http://www.polytech.univ-smb.fr/index.php?id=2884&L=1">LOCIE UMR 5271</a>) laboratory. They are both devoted to the problems of the Heat And Mass (HAM) transfer problems in porous materials. More specifically, the first publication is devoted to the problems of explicit/implicit time discretizations and how to overcome slightly the CFL-type stability limit:</p>
<ul>
<li>S. Gasparin, J. Berger, <strong>D. Dutykh</strong> & N. Mendes. <a href="https://hal.archives-ouvertes.fr/hal-01404578/"><em>Numerical schemes for the solution of non-linear moisture transfer in porous materials: implicit or explicit? That is the question!</em></a> Submitted, 2016</li>
</ul>
<p>The second manuscript is about the spatial discretization of HAM transfer equations using Scharfetter-Gummel-type schemes. Then, the proposed schemes are used to validate the model against experimental data:</p>
<ul>
<li>J. Berger, S. Gasparin, <strong>D. Dutykh</strong> & N. Mendes. <a href="https://hal.archives-ouvertes.fr/hal-01419018/"><em>How to accurately predict moisture front?</em></a> Submitted, 2016</li>
</ul>
<p>As always, these preprints are freely available to read and to download though the <a href="https://hal.archives-ouvertes.fr/">HAL</a> server.</p>
<p>I hope that these works will appear soon in the specialized referred journals.</p>
<p><strong>UPDATE:</strong> There are two available videos of short talks devoted to the topics covered in publications above:</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/https://youtu.be/Ab_nxWOIca8" frameborder="0" allowfullscreen=""></iframe>
<iframe width="560" height="315" src="https://www.youtube.com/embed/https://youtu.be/https://youtu.be/gUiJ0YYs-WE" frameborder="0" allowfullscreen=""></iframe>
<hr />
Peakons
2016-10-30T00:00:00+00:00
http://www.denys-dutykh.com//2016/10/30/Peakons
<p>I apologize for a long break from the last post. However, it does not mean that we did not do anything :) On the contrary!</p>
<p>The first preprint is devoted to the famous Whitham equation as a model of long capilllary-gravity waves. It is not available yet on <a href="https://arxiv.org/">Arxiv</a>, but it can be found, for example, in <a href="https://www.researchgate.net/publication/308512590_The_Whitham_equation_with_surface_tension">ResearchGate</a> or I can send it back after a simple request by e-mail:</p>
<ul>
<li>E. Dinvay, D. Moldabayev, <strong>D. Dutykh</strong> & H.Kalisch. <em>The Whitham equation with surface tension</em>, Submitted, 2016</li>
<li><em>Abstract</em>: The Whitham equation was proposed as an alternate model equation for the simplified description of unidirectional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we derive the Whitham equation from the Hamiltonian theory of surface water waves while taking into account surface tension. It is shown numerically that in various scaling regimes the Whitham equation gives a more accurate approximation of the free surface problem for the Euler system than other models like the KdV, BBM or Kawahara equation. Only in the case of very long waves with positive polarity do the KdV and Kawahara equations outperform the Whitham equation with surface tension.</li>
</ul>
<p>Another preprint was submitted a couple of days ago. It reports our recent findings of peakon-like travelling waves to capillary-gravity Serre-Green-Naghdi equations in the critical regime:</p>
<ul>
<li>D. Mitsotakis, <strong>D. Dutykh</strong>, A. Assylbekuly & D. Zhakebaev. <a href="https://hal.archives-ouvertes.fr/hal-01388481/">On weakly singular and fully nonlinear travelling shallow capillary-gravity waves in the critical regime</a>, Submitted, 2016</li>
<li><em>Abstract</em>: In this Letter we consider long capillary-gravity waves described by a fully nonlinear weakly dispersive model. First, using the phase space analysis methods we describe all possible types of localized travelling waves. Then, we especially focus on the critical regime, where the surface tension is exactly balanced by the gravity force. We show that our long wave model with a critical Bond number admits stable travelling wave solutions with a singular crest. These solutions are usually referred to in the literature as peakons or peaked solitary waves. They satisfy the usual speed-amplitude relation, which coincides with Scott-Russel’s empirical formula for solitary waves, while their decay rate is the same regardless their amplitude. Moreover, they can be of depression or elevation type independent of their speed. The dynamics of these solutions are studied as well.</li>
</ul>
<p>Let us hope they will be published soon!</p>
<hr />
Latest results
2016-07-05T00:00:00+00:00
http://www.denys-dutykh.com//2016/07/05/Summer
<p>In this post I would like to mention a couple of recent submissions that we prepared with my new and old co-authors.</p>
<p>First of all, with my new brazilian friends (Julien Berger and Nathan Mendes) we submitted a manuscript devoted to the understanding of some inverse problems in the heat conduction (yes, I am totally <em>open</em> to new topics). The preprint is already available on HAL server (it contained too many pictures to :</p>
<ul>
<li>J. Berger, <strong>D. Dutykh</strong> & N. Mendes. <a href="https://hal.archives-ouvertes.fr/hal-01334414">On the optimal experimental design for heat and moisture parameter estimation</a>, Submitted, 2016</li>
<li><em>Abstract</em>: In the context of estimating material properties of porous walls based on in-site measurements and identification method, this paper presents the concept of Optimal Experiment Design (OED). It aims at searching the best experimental conditions in terms of quantity and position of sensors and boundary conditions imposed to the material. These optimal conditions ensure to provide the maximum accuracy of the identification method and thus the estimated parameters. The search of the OED is done using the Fisher information matrix and a priori knowledge on the parameters. The methodology is applied for two cases. The first one deals with purely conductive heat transfer, while the second one combines a strong coupling between heat and moisture transfer.</li>
</ul>
<p>The second preprint is devoted to my more traditional topics, <em>i.e.</em> water wave modelling. Namely, we consider the deep water case and discuss several models, some of them being well-known and some new. The variational derivations and other averaging techniques are equally considered:</p>
<ul>
<li><strong>D. Dutykh</strong>, D. Clamond & M. Chhay. <a href="https://hal.archives-ouvertes.fr/hal-01340379">Serre-type equations in deep water</a>, Submitted, 2016</li>
<li><em>Abstract</em>: This manuscript is devoted to the modelling of water waves in the deep water regime with some emphasis on the underlying variational structures. The present article should be considered as a review of some existing models and modelling approaches even if new results are presented as well. Namely, we derive the deep water analogue of the celebrated Serre-Green-Naghdi equations which have become the standard model in shallow water environments. The relation to existing models is discussed. Moreover, the multi-symplectic structure of these equations is reported as well. The results of this work can be used to develop various types of robust structure-preserving variational integrators in deep water. The methodology of constructing approximate models presented in this study can be naturally extrapolated to other physical flow regimes as well.</li>
</ul>
<p>Now I think it is a good time to recharge the batteries… to undertake new research in September :)</p>
<p><img src="/public/pics/SmallRelax.png" alt="Summer" /></p>
<hr />
French-Spanish workshop
2016-05-14T00:00:00+00:00
http://www.denys-dutykh.com//2016/05/14/Workshop
<p><img src="/public/pics/Valladolid.jpg" alt="Valladolid" /></p>
<p>Tomorrow I departure for the next destination - Valladolid (Spain) for a French-Spanish Workshop on Evolution Problems (FSWEP16). This meeting will be hosted by <a href="http://imuva.uva.es/">imUVa</a> and <a href="http://www.uva.es/">Universidad de Valladolid</a>. The Programme can be seen <a href="http://imuva.uva.es/files/cartelfswep16.pdf">here</a>. I would like to thank the organizers for giving me an opportunity to speak there. My talk will be devoted to the derivation of Galilean invariant and energy-consistent long wave models.</p>
<hr />
Lectures at Curitiba and internal waves
2016-04-05T00:00:00+00:00
http://www.denys-dutykh.com//2016/04/05/Curitiba
<p>During these two weeks (1st – 14th of April 2016) I will be in <a href="https://en.wikipedia.org/wiki/Curitiba">Curitiba</a>, Brazil to deliver some lectures on the Numerical Analysis at a PhD school in <a href="http://www.pucpr.br/">PUCPR University</a>. I found on YouTube a short video presenting this University:</p>
<iframe width="480" height="360" src="http://www.youtube.com/embed/lvzgYOu3cHk" frameborder="0"> </iframe>
<p>The Lecture notes are already available. They will be probably updated in the nearest future to the latest version (including some minor corrections and perhaps some new material):</p>
<ul>
<li><strong>D. Dutykh</strong>. <a href="https://cel.archives-ouvertes.fr/cel-01256472/">A brief introduction to pseudo-spectral methods: application to diffusion problems</a>, Lecture Notes, 2016</li>
</ul>
<p>Otherwise, recently with Didier Clamond we submitted a new mauscript where we report the <a href="https://en.wikipedia.org/wiki/Multisymplectic_integrator">multi-symplectic</a> structure for the Green-Naghdi-type system describing long internal waves evolution. It seems that this system was derived for the first by E. Barthélémy (1989) in his PhD thesis done at the University of Grenoble. For simplicity, we adopt the rigid lid approximation, but the multi-symplectic structure can be generalized to the free surface case as well. The preprint is already freely available to download:</p>
<ul>
<li>D. Clamond, <strong>D. Dutykh</strong>. <a href="https://hal.archives-ouvertes.fr/hal-01296552/">Multi-symplectic structure of fully-nonlinear weakly-dispersive internal gravity waves</a>, Submitted, 2016</li>
</ul>
<p>Any comments and suggestions are welcome!</p>
<hr />
Singular solitary waves
2016-03-18T00:00:00+00:00
http://www.denys-dutykh.com//2016/03/18/SingularSolitaryWaves
<p><img src="/public/pics/PPA.png" alt="Phase plane analysis" /></p>
<p>Finally, yesterday with my collaborators from the <a href="http://unice.fr/">University of Nice</a> we submitted our manuscript on the construction of singular solitary wave solutions using the classical Phase Plane Analysis (PPA) and some methods from <em>effective</em> Algebraic Geometry.</p>
<p>This work was presented in many seminars from Japan (Yokohama) to Russia (Novosibirsk) and Austria (Linz). So, now you can find all the details about computations in our preprint available at all popular preprints servers:</p>
<ul>
<li><a href="http://math.unice.fr/~didierc/">D. Clamond</a>, <strong>D. Dutykh</strong>, <a href="http://math.unice.fr/~galligo/">A. Galligo</a>. <a href="https://hal.archives-ouvertes.fr/hal-01290471">Algebraic method for constructing singular steady solitary waves: A case study</a>, Submitted, 2016</li>
</ul>
<hr />
Quantum Chaos
2016-03-01T00:00:00+00:00
http://www.denys-dutykh.com//2016/03/01/QuantumChaos
<p><img src="/public/pics/Cardioid.png" alt="Quantum chaos" /></p>
<p>Recently I had to travel from Bucharest to Lyon (with a connecting flight at Istanbul). Arguably the best thing to do during a flight is to read. So, during my recent flights I read the following interesting article that I am pleased to recommend to you:</p>
<ul>
<li>M. Porter. <a href="http://arxiv.org/abs/nlin/0107039">An Introduction to Quantum Chaos</a>, <a href="http://arxiv.org/">Arxiv.org</a>, 2001</li>
</ul>
<p>The reference is a little bit long, but it includes a lot of interesting historical information. Finally, I can say that after reading it I have an impression to have understood something about Quantum Chaos (QC)! :) In particular, you will learn that QC appears in three different kinds. I also had an impression that this research topic is essentially open. Very little is known about genuine QC.</p>
<hr />
Gravity waves and A. Grothendieck
2016-02-14T00:00:00+00:00
http://www.denys-dutykh.com//2016/02/14/GravityWaves
<p><img src="/public/pics/GWaves.jpg" alt="Gravity waves" /></p>
<p>Currently there is a general excitement (both among the scientists and in mass media) about the detection of <em>gravity waves</em>. Certainly it is an important breakthrough in technology and also in Signal Processing (yes, signal processing is responsible at least for half of the detection success, the description of employed algorithms can be found <a href="http://statweb.stanford.edu/~candes/papers/ChirpDetection.pdf">here</a>). However, for me this event is another confirmation of a tremendous gap which exists between the theoretical and experimental physics today (and possibly until forever). The state of the art in experimental physics today (2016) corresponds to the state of the art of theoretical physics in 1916. Theoretical physicists work today on the structure of space and time on Planck’s scales (i.e. ~ 10^{-33} cm / 10^{-44} s). There are some reasons to think that most probably we shall never be able to access experimentally to these scales.</p>
<p>On a more positive side, I would like to share another article from Inference Review, devoted to <a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80_%D0%93%D1%80%D0%BE%D1%82%D0%B5%D0%BD%D0%B4%D1%96%D0%BA">Alexander Grothendieck</a>. This person for me is a symbol of the scientific revolution. The article is available both in French and in English for your convenience:</p>
<ul>
<li>
<p>P. Cartier. <a href="http://inference-review.com/article/un-pays-dont-on-ne-connaitrait-que-le-nom">Alexander Grothendieck. Un pays dont on ne connaîtrait que le nom</a>, <a href="http://inference-review.com/">Inference Review</a>, <strong>1</strong>(1), 2014</p>
</li>
<li>
<p>P. Cartier. <a href="http://inference-review.com/article/a-country-known-only-by-name">Alexander Grothendieck. A Country Known Only by Name</a>, <a href="http://inference-review.com/">Inference Review</a>, <strong>1</strong>(1), 2014</p>
</li>
</ul>
<p>Many fields of Mathematics (and of Science in general) wait for their A. Grothendieck!</p>
<hr />
An experimental approach to Mathematics
2016-02-09T00:00:00+00:00
http://www.denys-dutykh.com//2016/02/09/GregChaitin
<p>Today I read an interesting article that I am delighted to share with you. These thoughts are written by a prominent (Argentine-American) computer scientist - Gregory Chaitin. It is published by <a href="http://inference-review.com/">Inference Review</a>:</p>
<ul>
<li>G. Chaitin. <a href="http://inference-review.com/article/doing-mathematics-differently">Doing Mathematics Differently</a>, <a href="http://inference-review.com/">Inference Review</a>, <strong>2</strong>(1), 2016</li>
</ul>
<p>It is devoted to Author’s baby - the Algorithmic Information Theory (AIT). The issues discussed there should be of interest for any working mathematician. Perhaps, this is my favourite passage from the aforementioned article:</p>
<blockquote>
<p>What if we take Gödel incompleteness very seriously and throw away rigor? Suppose you have a property of the prime numbers which has been checked on the computer. You graph it and there is a beautiful curve, and it is fit beautifully by a very simple equation. What if you cannot prove it? A physicist would publish anyway. But a pure mathematician does not care how much empirical evidence there is, or how accurately this simple formula fits the curve. You need a proof!</p>
</blockquote>
<hr />
Full Euler solver on general bottoms
2016-02-09T00:00:00+00:00
http://www.denys-dutykh.com//2016/02/09/EulerCode
<p>Today I put on my <a href="https://github.com/dutykh/">repository</a> at Github the Matlab sources of a pseudo-spectral full Euler solver on general (<em>smooth</em>) bottoms in 2D. It is based on the method of dynamic conformal mappings proposed by Ovsyannikov in early 70’s. More details can be found at the following URL:</p>
<ul>
<li><a href="https://github.com/dutykh/Euler_bottom/">https://github.com/dutykh/Euler_bottom/</a></li>
</ul>
<hr />
On travelling waves in two-component systems
2016-01-30T00:00:00+00:00
http://www.denys-dutykh.com//2016/01/30/Delia
<p><img src="/public/pics/Sketch.png" alt="Sketch of the domain" /></p>
<p>A few days ago, with Delia Ionescu-Kruse (IMAR, Bucharest, Romania) we submitted a study of solitary and cnoidal wave solutions in three different two-component systems:</p>
<ul>
<li>The classical Green-Naghdi (GN) model</li>
<li>A new variant of GN equations (proposed by Delia a couple of years age)</li>
<li>The two-component Camassa-Holm equations</li>
</ul>
<p>In our study we describe all possible travelling gravity waves which may arise in that systems:</p>
<ul>
<li><strong>D. Dutykh</strong>, D. Ionescu-Kruse. <a href="https://hal.archives-ouvertes.fr/hal-01294603/">Travelling wave solutions for some two-component shallow water models</a>, Accepted to <a href="http://www.journals.elsevier.com/journal-of-differential-equations/">J. Diff. Equations</a>, 2016</li>
</ul>
<p>However, my dear co-author Delia strongly forbade me to share the preprint before it gets accepted to a peer-reviewed Journal. So, please, be patient to see these results.</p>
<hr />
Lecture notes on pseudo-spectral methods
2016-01-19T00:00:00+00:00
http://www.denys-dutykh.com//2016/01/19/LectNotes
<p>In this brief post I would like to advertise the notes that I made available recently. In these notes we attempt to introduce the reader to pseudo-spectral methods. The main applications include the heat conduction problems (i.e. parabolic PDEs) since initially the notes were intended for a PhD school on thermal modelling in buildings. The notes cover some other topics such as stochastic approaches (Feynman-Kac) and Trefftz methods which are put into Appendix. If you are interested, you can download the document at this URL address:</p>
<ul>
<li><strong>D. Dutykh</strong>. <a href="https://cel.archives-ouvertes.fr/cel-01256472/">A brief introduction to pseudo-spectral methods: application to diffusion problems</a>, <em>Lecture notes</em>, 38 pp, 2016</li>
</ul>
<hr />
Tsunami waves
2015-12-12T00:00:00+00:00
http://www.denys-dutykh.com//2015/12/12/Tsunami
<p><img src="/public/pics/Tsunami.jpg" alt="Tsunami" /></p>
<p>I would like to advertise here a general audience article devoted to tsunami waves, which was written by with a modest input from me. You can consult it here:</p>
<ul>
<li><strong>Tridib Banerjee</strong>, <a href="http://www.scrivial.com/articles/a-dive-into-tsunami-the-terrifying-energy-it-possesses/">A dive into tsunami - the terrifying energy it possesses!</a> <a href="http://www.scrivial.com/">Scrivial.com</a>, 2015</li>
</ul>
<p>–</p>
Energy conservation in dispersion improved models
2015-11-23T00:00:00+00:00
http://www.denys-dutykh.com//2015/11/23/mSGN
<p><img src="/public/pics/conserve-energy.gif" alt="Energy" /></p>
<p>Yesterday with my collaborators <a href="http://math.unice.fr/~didierc/">D. Clamond</a> and <a href="https://sites.google.com/site/dmitsot/">D. Mitsotakis</a> we submitted a new manuscript devoted to an improved version of Serre-Green-Naghdi equations. In particular, we improve its dispersion relation properties. Somebody will say that this point is not really new and he/she will be totally right. The novelty in our study consists in the fact that we improve the dispersion relation property, while preserving the energy conservation and having the Galilean invariance property. These two points are essential for physically sound modeling of water waves.</p>
<p>The details can be found in our preprint:</p>
<ul>
<li>D. Clamond, <strong>D. Dutykh</strong> & D. Mitsotakis. <a href="https://hal.archives-ouvertes.fr/hal-01232370">Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics</a>, Submitted, 2015</li>
</ul>
<hr />
Visit to IMAR
2015-11-14T00:00:00+00:00
http://www.denys-dutykh.com//2015/11/14/IMAR
<p><img src="/public/pics/IMAR.png" alt="IMAR" /></p>
<p>The next week (more precisely from the 16th to 21st of November) I am going to spend at the Institute of Mathematics ‘<em>Simion Stoilow</em>’ (<a href="http://www.imar.ro/">IMAR</a>), <a href="https://en.wikipedia.org/wiki/Bucharest">Bucharest</a>, <a href="https://en.wikipedia.org/wiki/Romania">Romania</a> after a kind invitation of Dr. <a href="http://imar.ro/organization/people/CVs/Ionescu-Kruse_Delia_CV.pdf">Delia Ionescu-Kruse</a>. I hope to be able to share with you the first results of this burgeoning collaboration.</p>
<hr />
Recent submissions
2015-11-07T00:00:00+00:00
http://www.denys-dutykh.com//2015/11/07/Submissions
<p><img src="/public/pics/Publications.jpg" alt="Publications" /></p>
<p>The last couple of weeks were very rich in terms of preprints submissions. No particular increase in productivity. It just happened that a certain number of research projects were finalized by this time. I don’t have any other explanation. In any case, here is the list of fresh manuscripts:</p>
<ul>
<li>
<p>M. Chhay, <strong>D. Dutykh</strong>, M. Gisclon & Ch. Ruyer-Quil. <a href="https://hal.archives-ouvertes.fr/hal-01224182/">Asymptotic heat transfer model in thin liquid films</a>, Submitted, 2015</p>
</li>
<li>
<p>G. Khakimzyanov & <strong>D. Dutykh</strong>. <a href="https://hal.archives-ouvertes.fr/hal-01223522/">On supraconvergence phenomenon for second order centered finite differences on non-uniform grids</a>, Submitted, 2015</p>
</li>
<li>
<p>G. Khakimzyanov, <strong>D. Dutykh</strong>, D. Mitsotakis & N. Shokina. <a href="https://hal.archives-ouvertes.fr/hal-01223510/">Numerical solution of conservation laws on moving grids</a>, Submitted, 2015</p>
</li>
<li>
<p>M. Chhay, <strong>D. Dutykh</strong> & D. Clamond. <a href="https://hal.archives-ouvertes.fr/hal-01221356/">On the multi-symplectic structure of the Serre-Green-Naghdi equations</a>, Submitted, 2015</p>
</li>
</ul>
<p>I am looking forward to read any comments from Referees!</p>
<hr />
Capillary-gravity waves
2015-10-25T00:00:00+00:00
http://www.denys-dutykh.com//2015/10/25/BabenkoCG2
<p><img src="/public/pics/Vvelocity.jpg" alt="Vertical velocity under a generalized solitary wave" /></p>
<p>A few days ago we (D. Clamond, A. Duran and me, Denys Dutykh) submitted a new manuscript devoted to the efficient computation of generalized capillary-gravity solitary waves in the full Euler equation with the free surface:</p>
<ul>
<li><strong>D. Dutykh</strong>, D. Clamond & A. Duran. <a href="https://hal.archives-ouvertes.fr/hal-01218989/">Efficient computation of capillary-gravity generalized solitary waves</a>. Submitted, 2015</li>
</ul>
<p>It is a follow-up of our previous study, which has just been accepted to the <a href="http://journals.cambridge.org/action/displayBackIssues?jid=flm">Journal of Fluid Mechanics</a>:</p>
<ul>
<li>D. Clamond, <strong>D. Dutykh</strong> & A. Duran. <a href="https://hal.archives-ouvertes.fr/hal-01081798/">A plethora of generalised solitary gravity-capillary water waves</a>. Acceptted to <a href="http://journals.cambridge.org/action/displayBackIssues?jid=flm">J. Fluid Mech.</a>, 2015</li>
</ul>
<p>In the new manuscript we explain in details the numerical methods employed to solve the nonlinear Babenko equation. The code produced after this study is freely available in open source at the following URL address:</p>
<ul>
<li><a href="https://github.com/dutykh/BabenkoCG/">https://github.com/dutykh/BabenkoCG/</a></li>
</ul>
<p>In particular, it allows to compute the physical quantities not only on the free surface, but also in the fluid bulk. The picture above shows the distribution of the vertical velocity component under a generalized solitary wave. You are invited to use it in your own research on the exciting topic of capillary-gravity waves!</p>
<hr />
Another visit to Novosibirsk
2015-09-30T00:00:00+00:00
http://www.denys-dutykh.com//2015/09/30/Nsk2
<p>From the 1st to 11th of October, 2015 I am going to visit for another time the <a href="http://www.ict.nsc.ru/">Institute of Computational Technologies</a> (“ICT SB RAS”). On the programme is the development of algorithms on adaptive grids and a wave/floating obstacle interaction problem. So, I hope to report on this research in the nearest future.</p>
<p><img src="/public/pics/ICT_small.jpg" alt="ICT SB RAS" /></p>
<hr />
SciCADE-2015
2015-09-12T00:00:00+00:00
http://www.denys-dutykh.com//2015/09/12/SciCADE
<p><img src="/public/pics/Potsdam.jpg" alt="Univ. Potsdam" /></p>
<p>Tomorrow I am going to Berlin, Germany in order to attend the “Scientific
Computation And Differential Equations” (<a href="http://scicade2015.math.uni-potsdam.de/scicade2015/">SciCADE-2015</a>) conference. Thanks to Professors Vassili Dougalis and Angel Duran for organizing a minisymposia devoted to nonlinear waves. I am going to speak there about the implicit ODEs and their applications to shallow capillary-gravity waves.</p>
<p><strong>Update:</strong> The slides of my talk can be downloaded <a href="http://www.denys-dutykh.com/innovaeditor/assets/admin/Talks/Dutykh-SciCADE-2015.pdf">here</a>.</p>
<hr />
Optimal hunting group size
2015-07-05T00:00:00+00:00
http://www.denys-dutykh.com//2015/07/05/Wolves
<p>As you probably already understood, my traditional research interests are focused on various questions stemming from the Hydrodynamics and free surface flow simulation. However, recently I opened a new research direction (at least for me) of mathematical modelling of animal behaviour. More precisely, we model the hunting process of a prey (<em>e.g.</em> an elk or a bison) by a group of wolves. It is quite well described by the laws of classical Mechanics.</p>
<p>In our first study we explain on the language of Dynamical Systems why there is an optimal hunting group size which maximizes the probability of prey’s capture. These results have been recently described in a succinct way in this manuscript:</p>
<ul>
<li>R. Escobedo, <strong>D. Dutykh</strong>, C. Muro, L. Spector & R.P. Coppinger. <em>Group Size Effect on the Success of Wolves Hunting</em>, Submitted, 2015</li>
</ul>
<p>I would like to express my grattitude to Ramond Escobedo who brought me into this true scientific adventure!</p>
<p><img src="/public/pics/Scheme.png" alt="Scheme" /></p>
<p>This is the summary of our first common work:</p>
<blockquote>
<p>Social foraging shows unexpected features such as the existence of an optimal group size above which additional individuals do not favor the success of the hunt. Previous work shows that the optimal group size is surprisingly small. In wolves hunting elk in Yellowstone Park, hunting success levels off beyond pack sizes of 4 individuals. This observation recently received support from a computational agent model which showed that the reduction of hunting success in large packs can be due to the emergence of privileged positions in the spatial wolf-pack formation. Subsequent observations of wolves hunting bison reinforce and document the hypothesis of the privileged positions. When hunting bison, the optimal wolf-pack size is between 9 and 13. We show here that this is in accordance with the computational model. Moreover, although the optimal group size is expected to be greater when hunting more dangerous prey, we show that this relation is surprisingly not linear: the computational model reveals that the optimal group size actually results from the opposite contributions of two critical distances separating wolves and prey. These distances strongly depend on the kind of prey, and can induce a different variation if a different prey is considered (<em>e.g.</em> moose).</p>
</blockquote>
<hr />
NumHyp-2015
2015-06-22T00:00:00+00:00
http://www.denys-dutykh.com//2015/06/22/Cortona
<p>I just came back from a wonderful workshop <a href="http://www.dmi.unict.it/NumHyp2015/">NumHyp-2015</a> devoted to the numerical aspects of hyperbolic equations. Thanks to the organizers (Giovanni Russo & Gabriella Puppo) for putting us together and creating a nice setting to exchange the ideas. It is probably the most beautiful room where I delivered a talk:</p>
<p><img src="/public/pics/Palazzone.jpg" alt="Il Palazzone" /></p>
<p>By the way, the slides of my talk can be already consulted on this URL:</p>
<ul>
<li>D. Dutykh. <a href="http://www.denys-dutykh.com/innovaeditor/assets/admin/Talks/Dutykh-NumHyp-2015.pdf">Modelling of shallow dispersive water waves</a>, 18 June 2015, <a href="https://en.wikipedia.org/wiki/Cortona/">Cortona</a>, Italy</li>
</ul>
<p>Tomorrow I am going to <a href="https://en.wikipedia.org/?title=Rouen">Rouen</a> (Upper Normandy, France) to visit my collaborator <a href="http://lmi.insa-rouen.fr/membres/10-membres/maitres-de-conference/20-caputo-jean-guy.html">Jean-Guy Caputo</a>. New results are coming.</p>
<hr />
sine-Gordon equation on graphs
2015-06-08T00:00:00+00:00
http://www.denys-dutykh.com//2015/06/08/sGraph
<p>Today we submitted a new preprint (in collaboration with <a href="http://lmi.insa-rouen.fr/membres/10-membres/maitres-de-conference/20-caputo-jean-guy.html/">Jean-Guy Caputo</a>) devoted to the dynamics of kink solutions of the celebrated <a href="http://en.wikipedia.org/wiki/Sine-Gordon_equation/">sine-Gordon equation</a> (sG). The particularity consists in the fact that the sG equation is posed on networks (<em>i.e.</em> graphs), which are not manifolds because of the junctions. This fact makes the formulation much trickier. The preprint can be already consulted at the HAL server:</p>
<ul>
<li><strong>D. Dutykh</strong> & J.-G. Caputo. <a href="https://hal.archives-ouvertes.fr/hal-01160840/">Discrete sine-Gordon dynamics on networks</a>. Submitted, 2015</li>
</ul>
<hr />
My favourites
2015-05-10T00:00:00+00:00
http://www.denys-dutykh.com//2015/05/10/Favourites
<p>This time I would like to share another text that I started to write recently. In this document I decided to assemble various sources of information (books, papers) and also some scientific software libraries which turned out to be extremely useful in my scientific work. I can say it in other words: my vision of some fields of (Applied) Mathematics has been strongly influenced by these references. The PDF file can be found at this public Github repository:</p>
<ul>
<li><a href="https://github.com/dutykh/libs/">https://github.com/dutykh/libs/</a></li>
</ul>
<p>The reader will notice that most of my favourite references are rather old. It can be mainly explained by the personal taste of the author.</p>
<p>Please, note that it is a work in progress. So, it should be considered only as a beginning of a longer story. From time to time I shall add new elements upon the arrival.</p>
<p>I hope you will find some inspiration in the books which inspired me some time ago. Have a nice reading and, of course, any comments/remarks/propositions are welcome!</p>
<p><img src="/public/pics/Library.jpg" alt="Library" /></p>
<hr />
Lecture notes
2015-04-21T00:00:00+00:00
http://www.denys-dutykh.com//2015/04/21/LectNotes
<p>With this post I would like to release the draft of my Lecture notes that I’ve been delivering recently. The PDF file can be downloaded here:</p>
<ul>
<li><a href="https://github.com/dutykh/hydro/">https://github.com/dutykh/hydro/</a></li>
</ul>
<p>I hope that even in the current preliminary version it might be already useful for the interested reader. Please, note that it is the <strong>work in progress</strong>. So, this file will be constantly evolving (almost on the daily basis). Consequently, the potential readers are invited to consult the corresponding GitHub repository for a newer version.</p>
<p>Any comments/remarks/propositions are welcome!</p>
<p>Have a nice reading!</p>
<p><img src="/public/pics/Hydrodynamics.jpg" alt="Hydrodynamics" />
<em>Copyright © Randal Ferman</em></p>
<hr />
Another visit to BCAM :)
2015-04-02T00:00:00+00:00
http://www.denys-dutykh.com//2015/04/02/Bcam2
<p>Tomorrow I am travelling again to the charming city of Bilbao (Bizkaia, Spain) after a kind invitation of <a href="http://www.bcamath.org/en/people/akhmatskaya/">Prof. Elena Akhmatskaya</a>. I am going to spend the whole month of April (and a bit of May) at the <a href="http://www.bcamath.org/en/">Basque Center for Applied Mathematics</a>. Besides numerous scientific collaborations I have with this center, this time I will also deliver a short introductory course on the Hydrodynamics in general, with a particular emphasis on the <a href="http://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics">Smoothed Particle Hydrodynamics</a> method.</p>
<p><img src="/public/pics/logo_bcam.jpg" alt="BCAM" /></p>
<p><img src="/public/pics/Bilbao.jpg" alt="Bilbao" /></p>
<hr />
Full Euler Matlab code
2015-03-12T00:00:00+00:00
http://www.denys-dutykh.com//2015/03/12/EulerDeep
<p>Recently I put on <a href="https://github.com/">GitHub</a> the <a href="http://www.mathworks.com/products/matlab/">Matlab</a> scripts of the full Euler solver on deep waters. The method is based on the conformal mapping technique along with the integration of linear terms. The source code can be accessed at the following URL address:</p>
<ul>
<li><a href="https://github.com/dutykh/ConformalEulerDeepWater/">https://github.com/dutykh/ConformalEulerDeepWater</a></li>
</ul>
<p>Special thanks go to <a href="https://www.researchgate.net/profile/Bernard_Ee2">Bernard Ee</a> and <a href="http://www.eng.tau.ac.il/~shemer/">Lev Shemer</a> from <a href="http://english.tau.ac.il/">Tel Aviv University</a> who helped me in initializing these computations with the <a href="http://en.wikipedia.org/wiki/Peregrine_soliton">Peregrine breather</a>.</p>
<p><img src="public/pics/PB_shot.png" alt="Peregrine breather" /></p>
<hr />
Water waves in bifurcating channels
2015-03-06T00:00:00+00:00
http://www.denys-dutykh.com//2015/03/06/NSWE-Junctions
<p>In collaboration with <a href="http://lmi.insa-rouen.fr/membres/10-membres/maitres-de-conference/20-caputo-jean-guy.html/">Jean-Guy Caputo</a> we continue our investigations on nonlinear waves interacting with branching channels. The previous work was devoted to the <a href="http://en.wikipedia.org/wiki/Sine-Gordon_equation/">sine-Gordon</a> kink dynamics, while nowadays we turn our attention to more classical waves. Namely, we consider the propagation of a solitary wave through a branching channel. Here are some preliminary simulations:</p>
<ul>
<li>Y-junction (with 90° angle)</li>
</ul>
<iframe width="480" height="360" src="http://www.youtube.com/embed/JJTeSdLTYUQ" frameborder="0"> </iframe>
<ul>
<li>T-junction (with 180° angle)</li>
</ul>
<iframe width="480" height="360" src="http://www.youtube.com/embed/gifKs4nLq3Y" frameborder="0"> </iframe>
<p>These numerical results were obtained by solving the classical <a href="http://en.wikipedia.org/wiki/Shallow_water_equations">Nonlinear Shallow Water Equations</a> (NSWE) on an unstructured triangular grid using the finite volume method, along with a time-adaptive scheme, of course.</p>
<p>Hope to post some further results soon!</p>
<p><strong>UPDATE (06/03/2015):</strong></p>
<p>There is also an available 3D animation for the Y-junction case with longer simulation time showing several reflections inside the channel (<em>please, don’t hesitate to open the window to the full screen</em>):</p>
<iframe width="480" height="360" src="http://www.youtube.com/embed/2L6HmZsiNqg" frameborder="0"> </iframe>
<hr />
Thermal effects in thin films
2015-03-04T00:00:00+00:00
http://www.denys-dutykh.com//2015/03/04/Thermal
<p>We continue to work on the dynamics of thin liquid films. In particular, two my colleagues from the <a href="https://www.univ-smb.fr/">University of Savoie Mont Blanc</a> - <a href="http://marx.chhay.free.fr/">Marx Chhay</a> and <a href="http://www.lama.univ-savoie.fr/~gisclon/">Marguerite Gisclon</a> have recently taken into account thermal effects in the thin film dynamics. A numerical simulation of the coupled model is shown here (<em>please, don’t hesitate to open the window on the full screen</em>):</p>
<iframe width="420" height="315" src="//www.youtube.com/embed/BDX07P3MlUo" frameborder="0"></iframe>
<p>The system evolution starts from a slightly perturbed free surface and a constant temperature distribution. Around the time instance t = 30 <em>s</em> it arrives to a permanent shape composed of a finite number of periodic travelling waves. The numerical scheme is based on the <a href="http://grid.engin.umich.edu/~gtoth/SWMF/GM/BATSRUS/Doc/HTML/DESIGN/node22.html">Rusanov numerical flux</a> and central finite differences for capillary and other source terms. The boundary conditions are periodic.</p>
<hr />
Thin film dynamics
2015-02-13T00:00:00+00:00
http://www.denys-dutykh.com//2015/02/13/ThinFilms
<p>I am getting also interested in the dynamics of thin films which flow down the vertical wall. There are many differences with the classical water wave models, besides flipping the wave propagation direction (<strong>horizontal</strong> to <strong>vertical</strong>). For instance, the viscosity plays a very important role in thin films, while it is negligible in water wave modelling. Another point is the surface tension force which has to be taken into account in the model. Below we show a numerical simulation of the instability development from a periodic perturbation (<em>please, don’t hesitate to open to the full screen</em>):</p>
<iframe width="480" height="360" src="http://www.youtube.com/embed/jkZ96ZmNAns" frameborder="0"> </iframe>
<p>We solve the so-called Shkadov’s model (also known as the viscid Saint-Venant equations) in a periodic domain using the finite volume method (Rusanov flux is used). The dispersive term is discretized using the central finite differences. The semi-discrete model is solved in time with an adaptive Runge-Kutta scheme.</p>
<p>A research direction to be continued!</p>
<hr />
Visit to BCAM
2015-01-31T00:00:00+00:00
http://www.denys-dutykh.com//2015/01/31/BCAM1
<p>I am going to spend the whole month of February, 2015 in the vibrating city of Bilbao (Bizkaia, Spain). The scientific reason for this trip is a kind invitation of <a href="http://www.bcamath.org/en/people/zuazua">Prof. Enrique Zuazua</a> to stay and participate in the life of the <a href="http://www.bcamath.org/en/">Basque Center for Applied Mathematics</a>. This visit promises to be rather productive, scientifically speaking.</p>
<p><img src="/public/pics/logo_bcam.jpg" alt="BCAM" /></p>
<p><img src="/public/pics/Bilbao.jpg" alt="Bilbao" /></p>
<hr />
Simone's visit
2015-01-19T00:00:00+00:00
http://www.denys-dutykh.com//2015/01/19/Simone
<p>This week I am pleased to host <a href="http://www.bcamath.org/en/people/srusconi/">Simone Rusconi</a> who prepares currently a Ph-D thesis at <a href="http://www.bcamath.org/en/">BCAM</a> under the supervision of <a href="http://www.bcamath.org/en/people/akhmatskaya/">Prof. Elena Akhmatskaya</a>. For me this visit represents the opening of a new research direction related to the material science and polymers modelling. I am really looking forward to announce the first results of this collaborative work.</p>
<hr />
Visit to Almaty, KZ
2014-11-29T00:00:00+00:00
http://www.denys-dutykh.com//2014/11/29/KZ
<p>Tomorrow I am travelling to Almaty, Kazakhstan. Perhaps, the destination is quite exotic, but I am grateful to Professor <a href="http://www.kaznu.kz/en/12625/page/Departments/Faculty_of_Mechanics_and_Mathematics_/Departments/Department_of_Mathematical_and_Computer_Modeling">Dauren Zhakebaev</a> for this kind invitation. In the programme ten lectures on Fluid Mechanics at <a href="http://www.kaznu.kz/en/">Al-Farabi Kazakh National University</a> and many exciting meetings. Looking forward to discover this country and the surrounding mountains!</p>
<p><img src="/public/pics/AlFarabi.jpg" alt="KazNU" /></p>
<hr />
Quasi-solitons in finite depth
2014-11-25T00:00:00+00:00
http://www.denys-dutykh.com//2014/11/25/quasisoliton
<p>With my collaborators from the <a href="http://www.iop.kiev.ua/index_en.php">Institute of Physics</a> (Kiev, Ukraine) we studied a particular high-order NLS-type equation valid in the range of intermediate waters (1.0 < kh < 5.0). It was derived by Yu. Sedletsky in 2003. Here you can watch a couple of videos showing the forward and backford propagation of a quasi-soliton in this system:</p>
<iframe width="480" height="360" src="http://www.youtube.com/embed/IYjAefqFz5E" frameborder="0"> </iframe>
<iframe width="480" height="360" src="http://www.youtube.com/embed/KafPGNbPMtI" frameborder="0"> </iframe>
<p>More information can be found in the final version of our manuscript devoted to these questions:</p>
<ul>
<li>I. Gandzha, Yu. Sedletsky & <strong>D. Dutykh</strong>. <a href="https://hal.archives-ouvertes.fr/hal-01084747/">High-order nonlinear Schrödinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions</a>. Accepted to <a href="http://ujp.bitp.kiev.ua/">Ukr. J. Phys.</a>, 2014</li>
</ul>
<hr />
New run-up algorithm
2014-11-20T00:00:00+00:00
http://www.denys-dutykh.com//2014/11/20/Runup
<p>Today we submitted a manuscript which describes a novel run-up algorithm to handle the moving shoreline. It is based on two main ingredients: the local analytic expansions put forward by a prominent mathematician <a href="http://en.wikipedia.org/wiki/Sofia_Kovalevskaya">Sofya Kovalevskaya</a> and on an analogy between the wave run-up problem with some problems in the compressible gas dynamics. Namely, we noticed that the wave propagation on a dry shore is similar to the gas outflow into the vacuum. So, the methods can be transposed from one field into the other provided the mathematical structure is similar. More information on these results and on the actual algorithm can be found in the following preprint available on the <a href="https://hal.archives-ouvertes.fr/">HAL</a> server:</p>
<ul>
<li>G. Khakimzyanov, N.Yu. Shokina, <strong>D. Dutykh</strong> & D. Mitsotakis. <a href="https://hal.archives-ouvertes.fr/hal-01084811">A new run-up algorithm based on local higher order analytic expansions</a>. Submitted, 2014</li>
</ul>
<hr />
New capillary-gravity waves
2014-11-19T00:00:00+00:00
http://www.denys-dutykh.com//2014/11/19/BabenkoCG
<p>We developed a new formulation, but also a new algorithm (based specifically on this formulation) to compute fully nonlinear generalized solitary capillary-gravity waves. Namely, the conformal mapping technique is employed to flatten the domain. In this way, the full Euler equations can be recast for steady solutions in the form of the so-called <em>Babenko equation</em>. Then, this equation written on the upper boundary of the transformed space is solved numerically using the Levenberg-Marquardt algorithm. By making various initial guesses we can obtain various solutions. Moreover, the conformal theory allows to obtain the complete information on the solution even inside the fluid domain. Here for the sake of illustration we show, for example, the vertical velocity distribution under a generalized solitary wave:</p>
<p><img src="/public/pics/Vvelocity.jpg" alt="Vertical velocity under a generalized solitary wave" /></p>
<p>More examples can be found in our preprint which has just been submitted:</p>
<ul>
<li>D. Clamond, <strong>D. Dutykh</strong> & A. Duran. <a href="https://hal.archives-ouvertes.fr/hal-01081798/">A plethora of generalised solitary gravity-capillary water waves</a>. Submitted, 2014</li>
</ul>
<hr />
Resonance enhancement
2014-11-04T00:00:00+00:00
http://www.denys-dutykh.com//2014/11/04/Resonances
<p>We all learnt in the school or university’ classes of Physics that the maximal amplitudes occur when a dynamical system enters into a resonance. Actually the amplitude will be infinite for linear systems. For all other conditions the amplitudes will be lower than in the exact resonance. This idealistic picture comes mainly from the asymptotic analysis of linear resonances in linear (or weakly nonlinear) systems. However, recently we showed that the picture is more complex for nonlinear resonances. In fact, the introduction of a small frequency detuning can enhance the resonance! For more details, please have a look on this recent preprint:</p>
<ul>
<li><strong>D. Dutykh</strong> & E. Tobisch. <a href="https://hal.archives-ouvertes.fr/hal-01079764">Resonance enhancement by suitably chosen frequency detuning</a>. Submitted, 2014</li>
</ul>
<hr />
Visit to Novosibirsk
2014-10-28T00:00:00+00:00
http://www.denys-dutykh.com//2014/10/28/Nsk
<p>During the period of 19 - 26th of October 2014 I visited Novosibirsk (“Akademgorodok”) and more specifically the <a href="http://www.ict.nsc.ru/">Institute of Computational Technologies</a> (“ICT SB RAS”). During my visit I delivered four talks at various institutes and seminars. For example, it was an enormous pleasure to speak at the former L.V. Ovsyannikov’s seminar at the Laboratory of Differential equations in the <a href="http://www.hydro.nsc.ru/">Lavrentyev Institute of Hydrodynamics</a>. Some video records can be found <a href="https://sites.google.com/site/zakonysohraneniaiinvarianty/zasedania-v-2014-godu">here</a>. Sorry, the talks were delivered in Russian.</p>
<p>Perhaps, it was one of the most intense visits I had recently. I would like to thank all the people who participated directly or indirectly to the organization of my stay there. Despite the cold weather, the visit was warm thanks to people <strong>warmth</strong>.</p>
<p><img src="/public/pics/ICT_small.jpg" alt="ICT SB RAS" /></p>
<hr />
Dynamic energy cascade in gKdV
2014-10-08T00:00:00+00:00
http://www.denys-dutykh.com//2014/10/08/gKdV
<p>Recently in collaboration with <a href="http://www.dynamics-approx.jku.at/lena/">Elena Tobisch</a> (Kartashova) we submitted already our third common work on the observation of the dynamic energy cascades in nonlinear dipersive PDEs. This time our study was devoted to the generalized KdV-type equations (up to <em>quintic</em> nonlinearities).</p>
<p>It is known that the classical (quadratic) KdV equation is modulationally stable, while the mKdV (with the cubic nonlinearity) is unstable if the nonlinear term sign coincides with the sign of the dispersive term. Quartic gKdV does not seem to have the Modulational Instability (MI), while the quintic gKdV shows MI for sufficiently large initial data. Our study is devoted to the interplay between MI effect on the dynamic energy cascade formation.</p>
<p>The preprint can be downloaded from the <a href="http://hal.archives-ouvertes.fr/">HAL</a> server:</p>
<ul>
<li><strong>D. Dutykh</strong>, E. Tobisch. <a href="http://hal.archives-ouvertes.fr/hal-01070799">Formation of the dynamic energy cascades in quartic and quintic generalized KdV equations</a>, Submitted, 2014</li>
</ul>
<p>For the sake of illustration here is a picture of the direct cascade in MI-unstable mKdV equation:
<img src="/public/pics/gKdV_T_160_10Modes.jpg" alt="Direct cascade" /></p>
<hr />
Visit to Linz
2014-09-14T00:00:00+00:00
http://www.denys-dutykh.com//2014/09/14/Linz
<p>The next two months I am going to spend at the <a href="http://www.jku.at/analysis/content/">Institut für Analysis</a>, <a href="http://www.jku.at/content">Johannes Kepler Universität Linz</a>, Austria after the kind invitation of Prof. <a href="http://www.dynamics-approx.jku.at/lena/">Elena Tobisch</a>. One of the main goals of this visit is to edit a contributed volume entitled <em>“New Approaches to Nonlinear Waves”</em>, which will be published by <a href="http://www.springer.com/">Springer</a> series <a href="http://www.springer.com/series/5304/">Lecture Notes in Physics</a>.</p>
<p><img src="/public/pics/LinzSciPark.jpg" alt="JKU Linz Science Park" /></p>
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Matlab + GPU
2014-08-21T00:00:00+00:00
http://www.denys-dutykh.com//2014/08/21/MatlabGPU
<p>In order to warm up before the Fall 2014, I decided to test Matlab GPU computing capabilities on my laptop which is equiped with the <a href="http://www.geforce.com/hardware/notebook-gpus/geforce-gtx-680m">GeForce GTX 680M</a> graphics card.</p>
<p>In order to take the full advantage of this hardware, the algorithm has to be suitable for the extreme parallelisation. That is why the choice of pseudo-spectral codes was natural. I have to say that porting of an existent Matlab (CPU) code to GPU is relatively easy.</p>
<p>Here you can see two simulations of the 2D incompressible <a href="http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations">Navier-Stokes</a> and <a href="http://en.wikipedia.org/wiki/Hasegawa%E2%80%93Mima_equation">Charney-Hasegawa-Mima</a> (CHM) equations. A random distribution of the vorticity was taken as the initial condition. No forcing. No dissipation in CHM. Just freely decaying turbulence. The resolution is 512x512 in both cases.</p>
<ul>
<li>Navier-Stokes simulation (Re = 10 000):</li>
</ul>
<iframe width="420" height="315" src="//www.youtube.com/embed/rKQ5nUTJKjU" frameborder="0"></iframe>
<ul>
<li>CHM simulation:</li>
</ul>
<iframe width="420" height="315" src="//www.youtube.com/embed/BP4VhEKSDF4" frameborder="0"></iframe>
<p>Now one can ask the legitimate question about the observed speed-ups. In my case (on the resolutions reported above) I observed the speed-up from 2 to 3 times. For higher resolutions this number is expected to increase. Note however, that the resolution is limited by the available GPU memory (~3.9 Gb for the GPU I have).</p>
<p>This modest speed-up can be explained by the following reasons:</p>
<ul>
<li>the GPU code is not yet fully optimized</li>
<li>the gain in speed is annihilated by the cost of memory transfer to/from GPU</li>
</ul>
<p>The profiling results seem to favour the latter explanation.</p>
<hr />
Whitham equation
2014-07-14T00:00:00+00:00
http://www.denys-dutykh.com//2014/07/14/WhithamEq
<p>Recently in collaboration with <a href="http://www.uib.no/persons/Daulet.Moldabayev">Daulet Moldabayev</a> and <a href="http://folk.uib.no/hka002/">Henrik Kalisch</a> (from the <a href="http://www.uib.no/en/math">Department of Mathematics</a>, <a href="http://www.uib.no/en">University of Bergen</a>) we performed a study on the relevance of the Whitham equation (which enjoys the <em>exact linear dispersion relation</em>) as a model for water waves (unidirectional propagation). Namely, the derivation of this equation from the Hamiltonian formalism was performed and the relevant parameters controlling the accuracy of this approximation were highlighted. Finally, this study is completed by performing comparisons in various prameters regimes with the full Euler equations solved by the dynamic conformal mapping technique.</p>
<p>This preprint can be accessed through the <a href="http://arxiv.org/">arxiv.org</a> server:</p>
<ul>
<li>D. Moldabayev, H. Kalisch & <strong>D. Dutykh</strong>. <a href="http://arxiv.org/abs/1410.8299">The Whitham Equation as a Model for Surface Water Waves</a>. Submitted, 2014</li>
</ul>
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Visit to Japan
2014-06-17T00:00:00+00:00
http://www.denys-dutykh.com//2014/06/17/RIMS
<p>Thanks to the kind invitation of Professor <a href="http://www.math.keio.ac.jp/~iguchi/index-e.html">Tatsuo Iguchi</a> I am going to visit Japan in July 2014. The scientific programme of this visit includes the following events:</p>
<ul>
<li><a href="http://www.kurims.kyoto-u.ac.jp/en/index.html">RIMS</a> Workshop: ‘‘<em><a href="http://www.math.keio.ac.jp/~iguchi/RIMS/">Mathematical Analysis in Fluid and Gas Dynamics</a></em>’’, July 2-4 at Kyoto University, with a talk on variational methods for water wave modeling</li>
<li>Nonlinear Analysis Seminar at Keio Univerity on July 8, where I am going to speak about shallow water waves with surface tension effects.</li>
</ul>
<p>Hope to see you soon in Japan!</p>
<hr />
Serre's pseudo-spectral solver
2014-06-03T00:00:00+00:00
http://www.denys-dutykh.com//2014/06/03/SerreGravity
<p>I just released a pseudo-spectral solver for the Serre-Green-Naghdi equations on the flat bottom. The scheme used in this Matlab script is described in the following publication:</p>
<ul>
<li><strong>D. Dutykh</strong>, D. Clamond, P. Milewski & D. Mitsotakis. <a href="http://hal.archives-ouvertes.fr/hal-00587994/"><em>Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations</em></a>, European Journal of Applied Mathematics, <strong>24</strong>(5), 761-787, 2013</li>
</ul>
<p>The script can be downloaded here:</p>
<ul>
<li><a href="https://github.com/dutykh/SerreGravityWave/">https://github.com/dutykh/SerreGravityWave/</a></li>
</ul>
<p><strong>Any comments are welcome!</strong></p>
<hr />
Kink in a Y-junction
2014-05-20T00:00:00+00:00
http://www.denys-dutykh.com//2014/05/20/Yjunction
<p>I would like to share another couple of simulations which are related to the previous post on the sine-Gordon equation. This time the geometry is a Y-junction (in contrast to T-junction shown below). The initial conditions are the same. These computations show one more time that the <em>qualitative</em> behaviour of the solution does not depend on the <em>angle</em> between the branches (but it depends on their <strong>width</strong>).</p>
<ul>
<li>Sub-critical kink (u = 0.75 < 0.94) reflected at the junction:</li>
</ul>
<iframe width="480" height="360" src="http://www.youtube.com/embed/TndqVetiAb0" frameborder="0"> </iframe>
<ul>
<li>Super-critical kink (u = 0.96 > 0.94) passing through the junction:</li>
</ul>
<iframe width="480" height="360" src="http://www.youtube.com/embed/eDAuJwAyGVY" frameborder="0"> </iframe>
<p><strong>Any comments are welcome!</strong></p>
<hr />
sine-Gordon equation
2014-05-19T00:00:00+00:00
http://www.denys-dutykh.com//2014/05/19/sineGordonFreeFem
<p>Thanks to my collaborator from <a href="http://en.wikipedia.org/wiki/Rouen">Rouen</a> (France) - <a href="http://lmi.insa-rouen.fr/membres/20-caputo.html">Jean-Guy Caputo</a> I discovered many interesting features of the <a href="http://en.wikipedia.org/wiki/Sine%E2%80%93Gordon_equation">sine-Gordon</a> equation. In particular, he invited me to study the dynamics of some special solutions (<em>kinks</em> and <em>breathers</em>) in tree-like geometries.</p>
<p>For the sake of illustration, here is a simulation of a kink passing through a T-junction, since it possesses enough energy to do it:</p>
<iframe width="480" height="360" src="http://www.youtube.com/embed/2pLqAKOrZHU" frameborder="0"> </iframe>
<p>And another kink which has a <em>sub-critical</em> velocity. Thus, it cannot pass through the junction:</p>
<iframe width="480" height="360" src="http://www.youtube.com/embed/Uz3gWlkg8_g" frameborder="0"> </iframe>
<p>More details on the obtained results can be found in our first common preprint:</p>
<ul>
<li>J.-G. Caputo & <strong>D. Dutykh</strong>. <a href="http://hal.archives-ouvertes.fr/hal-00951705/">Nonlinear waves in networks: a simple approach using the sine-Gordon equation</a>. Submitted, 2014</li>
</ul>
<p>Moreover, I just released in Open source the script for <a href="http://www.freefem.org/ff++/">FreeFem++</a> that we used to perform the simulations shown above:</p>
<ul>
<li><a href="https://github.com/dutykh/sineGordon_FreeFem/">https://github.com/dutykh/sineGordon_FreeFem</a></li>
</ul>
<p><strong>Any comments are welcome!</strong></p>
<hr />
Dynamic energy cascades
2014-05-15T00:00:00+00:00
http://www.denys-dutykh.com//2014/05/15/D-cascade
<p>Recently in collaboration with <a href="http://www.dynamics-approx.jku.at/lena/">Elena Tobisch</a> (Kartashova) we observed a direct energy cascade in the <strong>modified</strong> Korteweg-de Vries equation (the modification consists in increasing the degree of nonlinearity). This cascade seems to be relatively universal and it should be observed in many nonlinear wave equations exhibiting the <a href="http://en.wikipedia.org/wiki/Modulational_instability">Modulational Instability</a>. The explanation of the physical mechanism along with direct numerical simulations can be found in our first common manuscript:</p>
<ul>
<li><strong>D. Dutykh</strong>, E. Tobisch. <a href="http://hal.archives-ouvertes.fr/hal-00990724">Direct dynamical energy cascade in the modified KdV equation</a>, Submitted, 2014</li>
</ul>
<p>Later on we were able to observe also the inverse and direct energy cascades simultaneously, which is represented on this image:</p>
<p><img src="/public/pics/MI_DI_T840.jpg" alt="Inverse cascade" /></p>
<p>More details on the inverse cascade properties can be found in this recent preprint:</p>
<ul>
<li><strong>D. Dutykh</strong>, E. Tobisch. <a href="http://hal.archives-ouvertes.fr/hal-00991944">Observation of the Inverse Energy Cascade in the modified Korteweg-de Vries Equation</a>, Submitted, 2014</li>
</ul>
<hr />
Matlab scripts
2014-05-13T00:00:00+00:00
http://www.denys-dutykh.com//2014/05/13/github-scripts
<p>Under the influence of some ideas about the <a href="http://www.openscience.org/blog/">open science</a> and <a href="http://www.siam.org/news/news.php?id=2064&goback=.gde_112393_member_232769759">reproducible research</a>, I decided to start to release at <a href="https://github.com/">GitHub</a> some code which could be useful for other researchers.</p>
<p>Here you can find two first releases containing the Matlab scripts that I previously used in my publications:</p>
<ul>
<li><a href="https://github.com/dutykh/Okada">Okada solution</a> implementation in Matlab to compute the static co-seismic displacements after an earthquake. This solution is widely used in <em>tsunami generation</em> problems</li>
<li><a href="https://github.com/dutykh/BabenkoSolitaryWave">Babenko equation</a> solver in order to compute solitary wave solutions to the full Euler equations with the free surface</li>
</ul>
<p>Some more codes will be released in the future. All my public repositories are regrouped under this URL address:</p>
<ul>
<li><a href="https://github.com/dutykh">https://github.com/dutykh</a></li>
</ul>
<hr />
Cnoidal waves stability
2014-05-12T00:00:00+00:00
http://www.denys-dutykh.com//2014/05/12/dsw-serre-with-johncarter
<p>I am very pleased to announce the release of our detailed study on the dynamics of various travelling wave-type solutions in the Serre-Green-Naghdi (SGN) system. This work was done in collaboration with <a href="https://sites.google.com/site/dmitsot/">Dimitrios Mitsotakis</a> and <a href="http://fac-staff.seattleu.edu/carterj1/web/">John D. Carter</a>.</p>
<p>In particular, we investigated the nonlinear dynamics of unstable (periodic) <em>cnoidal waves</em> in the SGN equations. It was shown that the growth rate measured in our experiments is in very good agreement with the previous purely <a href="http://www.sciencedirect.com/science/article/pii/S0997754610001287">linear spectral analysis</a>. Potentially, this mechanism could be responsible of the periodic wave train amplification in shallow waters.</p>
<p>You can find more details on the obtained results in our preprint available at the <a href="http://hal.archives-ouvertes.fr/">HAL</a> server:</p>
<ul>
<li>D. Mitsotakis, <strong>D. Dutykh</strong>, J.D. Carter. <a href="http://hal.archives-ouvertes.fr/hal-00984035">On the nonlinear dynamics of the traveling-wave solutions of the Serre equations</a>, Submitted, 2014</li>
</ul>
<hr />
Start of the blog
2014-05-11T00:00:00+00:00
http://www.denys-dutykh.com//2014/05/11/try2
<p>This is going to be my second try to have a blog with some information about my research activitied and some other new developments in my fields of interest:</p>
<ul>
<li>Hydrodynamics / <a href="http://en.wikipedia.org/wiki/Fluid_dynamics">Fluid dynamics</a></li>
<li><a href="http://en.wikipedia.org/wiki/Numerical_analysis">Numerical methods</a> and <a href="http://en.wikipedia.org/wiki/Computational_science">scientific computing</a></li>
<li>More generally the <a href="http://en.wikipedia.org/wiki/Applied_mathematics">Applied mathematics</a></li>
</ul>
<hr />