Denys DUTYKH's Blog of an applied mathematician

Fast and accurate computation of cnoidal waves

There are many points to mention and describe briefly today.

First of all, together with my collaborator Didier Clamond, we submitted a manuscript, which describes a new method for the computation of cnoidal waves in the full Euler equations with free surface:

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:

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.

In the continuation of our previous short publication on peaked solitary capillary-gravity waves, we submitted the full manuscript, which investigates this system of equations (capillary-gravity Serre model) in more details:

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.

Finally, the following week I am going to spend at Al-Farabi Kazakh National University

KazNU


Open source problems

I would like to share with you a pretty interesting talk given by William Stein, who is the creator and the main developer in the SageMath open source project. You can watch the video here:

The slides can be downloaded separately here.

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.

Just to give some idea about the content, I give here a couple of quotations from this talk: * “Every great open source math library is built on the ashes of someone’s academic career.” * “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.”

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.


Two interesting talks

I would like to share with you a couple of videos coming from the Michigan Engineering (University of Michigan) that I find particularly interesting and informative. The first one is given by Phil Roe, 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.

As a side remark, there is a pretty interesting report produced by NASA on a very close topic:

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):

Have a pleasant listening!


Heat and Mass Transfer (HAM)

Heat

I would like to share here two recent publications which resulted from my collaboration with one Brazilian (LST, PUCPR) and one French (LOCIE UMR 5271) 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:

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:

As always, these preprints are freely available to read and to download though the HAL server.

I hope that these works will appear soon in the specialized referred journals.

UPDATE: There are two available videos of short talks devoted to the topics covered in publications above:


Peakons

I apologize for a long break from the last post. However, it does not mean that we did not do anything :) On the contrary!

The first preprint is devoted to the famous Whitham equation as a model of long capilllary-gravity waves. It is not available yet on Arxiv, but it can be found, for example, in ResearchGate or I can send it back after a simple request by e-mail:

  • E. Dinvay, D. Moldabayev, D. Dutykh & H.Kalisch. The Whitham equation with surface tension, Submitted, 2016
  • Abstract: 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.

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:

  • D. Mitsotakis, D. Dutykh, A. Assylbekuly & D. Zhakebaev. On weakly singular and fully nonlinear travelling shallow capillary-gravity waves in the critical regime, Submitted, 2016
  • Abstract: 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.

Let us hope they will be published soon!