Interview with Eric Gourgoulhon

0:00 So, I'll leave it here. I'm just saying to it that it's about 3 in the afternoon of the 23rd of May, 2000, and I'm speaking with Eric Gordon. So, well, I guess I wanted to, I understand that your group here works on simulations of neutron star and neutron star binaries. Yes, yes. And one of the things I understand that distinguishes you from some of the numerical groups that I've talked to is that you use spectrum. Ah, yes, yes. all other groups are using mostly finding difference methods and all groups use spectrum methods so maybe you want to know why we chose these methods instead of finding difference in fact so one pretty historical thing it was before I joined this group in the middle of the 80s, then Jean-Anne-Marc and Silvano Bonanzola decided to begin numerical relative computation by spectral methods, because they knew that spectral methods are well used in hydrodynamics, meteorology, oceanography, and they are well developed in this field. But apparently they were not developed at all, not only in numerical relativity, but also in astrophysics in general. They were almost not employed, so we decided to employ them. And the basic idea is that the mathematical theory of these methods is well developed. There are many theorems about these methods. And also they are very, very precise for smooth fields. can especially this is usually especially for 3d computation where you are still computation nowadays are still limited by the capacity of computers yes in 3d and so you cannot put so many points in your grid and some not as many as you want so it is very satisfactory to have a precise method because you can use less points to get the same accuracy and also one of the

2:30 One nice feature of spectral methods, in my opinion, I think it is maybe the nicer feature, it is that you can use a singular coordinate system, such as spherical coordinates. Because in this case, the main idea of spectral method is to develop each field on the basis of functions and you work on the coefficient of these expansions and so when you are using a singular coordinate system such as spherical coordinates you can the singularity is inside your basis function and not in the coefficient when you expand your field so you have perfectly regular coefficient finite numbers and something so so it's good for computer and it is nice to be able to use spherical coordinates because most objects that we're studying are stars or black holes, so they have a spherical topology. So, obviously, spherical coordinates are well adapted to this object. And I know, by discussion with colleagues from other groups, that it is very difficult to use spherical coordinates in 3D finite difference codes. So, most people use Cartesian finite difference. And so, this is a problem of the boundary conditions especially for black holes problem you it is a big one of the big problem of nowadays computation of binary black holes is to put correct boundary condition on the horizon and since the horizon is a spherical topology and since you are using cartesian coordinates you have some trouble in describing this so it introduced some numerical noise and and this is a source of trouble. So, also, when you describe neutron star, if you do the hydrodynamics of a neutron star, you have to put some boundary condition on the velocity field at the surface of a star, and this is also, you have also trouble to put this boundary condition with Cartesian coordinates. It's much more easier to use spherical coordinates, of course. So, we are using systematically spherical coordinates, And when we have a binary, then we use two spherical coordinate systems, one centered on each star, so that each object is more or less spherical in one coordinate system. Of course, it would be very bad to use only one coordinate system centered, let's say, the center of mass of the system, because it is not well adapted to describing two objects like this.

5:00 So, we are using two spherical coordinate systems, and we are also using a coordinate transformation, so that in our coordinates, let's say r prime, theta prime, phi prime, the surface of a star is exactly r prime equal to 1. So, it means even if it is highly distorted, we are mapping the star to a sphere, in fact. With this, we can put exact boundary conditions with the surface of the stars. And also, one of the nice features, as far as numerical relativity is concerned, one of the nice features of spherical coordinates is that you can identify easily or single out the asymptotic behavior of your fields, of the metric fields, because it corresponds to R equal infinity, so you have only one coordinate to deal with, not X, Y, Z, or something like that. and also we are systematically using multi-domain in fact spectral methods we are not using one domain we are using multi-domain and the most external domain is compactified so that we can go up to spatial infinity and this is very interesting because this is the only way when you know the metric by advance it is flat space-time so you know the exact boundary condition to put with you put simply the Minkowski metric works well. We did this for single rotating neutron stars, and now we are doing this for binary neutron stars, and it is a nice feature. So, just to sum up, I would say that one of the most interesting features of the spectrometer is to allow the use of spherical coordinates, besides the precision, of course, you have also spherical coordinates. So, my understanding is that with the spectral method, with the finite differencing, people have this grid, and at the various points on the grid, you define the values of whatever function you're interested in, but in the spectral method, you actually don't have a grid at all? We do have a grid because this is what we call collocation point in the terminology of the spectral methods.

7:30 So it means that given a field, you can describe it either by its value of the collocation point or either by the coefficient of its expansion, the basis function. And you can go from one representation to another by fast Fourier transforms or fast Chebyshev transforms So you can, so basically for linear operation, we do it in the coefficient space. This is the same as for Fourier series, after all, Fourier series are a particular case of spectral methods. So you can describe either function by its coefficient of its Fourier expansion, or either by the values and points on the circle. And so for all linear operations, such as taking the gradient of a field, the vector field, solving the Poisson equation, all these things are performed on the coefficient. So it is matrix operations on the coefficient. But for non-linear operation, and there are many non-linear terms in numerical relativity, thus we perform this non-linear operation in the physical space. We simply multiply point by point in the physical space, in the collocation point. So the final difference is that we cannot choose this collocation point as we want. They have to have a specific location which are linked to some zero of Chebyshev polynomials because for the radial part we use Chebyshev polynomials. and it means that it is not a uniform grid it is adapt to this so we have both representations but the grid isn't a sort of square no it is a sphere because we're using spherical coordinates so we have a grid in R theta phi In fact, since we're using a multi-domain, we have a multi-grid, if you want, so we have the inner grid is a topology of a ball, the waveosphere has a limit, and then we have grids, we have topology of shells around, and the most external shells is compactified so that it extends to spatial infinity.

10:00 So would it be possible to achieve the spherical symmetry by just having the grid be spherical and not use the spherical symmetry? You mean if you want to use a spherical, to compute something which is exactly spherical asymmetric? So in this case, this is also why it is not, spherical coordinates are nice, because you can take a special grid with only one point in phi and one point in theta, And then you have to take a few points, a few coefficients in R. Whereas if you try to treat this in Cartesian coordinates, of course, you have to take many points in X, Y, and Z. And so this is costly in terms of CPU time. And in our case, when we compute a spherical asymmetric star model, it is very fast because we have only one point in two directions. And we can do this, we did this, in fact to prepare our initial condition for binary in the trans-star we took a solution of OpenAimov-Volkov equations as initial condition to compute the equilibrium of this binary system and so it is exactly spherical so we compute like this and we put and then we perform some iterations because the equations are non-linear so we iterate to get the structure. with the full 3-D deformation of each star. What's the reason that you choose to use the combination of two methods to have a grid in the... No, it is not a combination. It is in the essence of this, what is called collocation spectrometer, or sometimes pseudo-spectrometer, is that you describe your field both on the collocation point or by its coefficient. So one of the, if you want, this is useful for non-linear operations because if it is simply a quadratic operation

12:30 like the product of two fields, you can do it also in coefficient space, convolution, the coefficient space, of course, but it is if it is more than quadratic non-linearity, take sine of field, I don't know, it's difficult to express on the coefficient of the development, so you have to do this in the spectral, in the coefficient points. For instance, such an example of such non-linearity is, you have the equation of state, you have complicated function of density to get the energy in the coefficient space you have to do it point by point given the question of state table the fact that our points are well chosen the spacing between the points is well determined means that you can use fast Fourier transform algorithm to compute from one point to the other and so this is why we're using such because it means that it is there is absolutely almost no cost in the code to go from one representation to another one. So we do not hesitate. When we need values at the physical point, we do it because it is very fast as compared with other parts of the code, as compared with the equation of set, or as compared with the resolution of some equations or some and also I should have stressed, I forgot among the advantage of using One is very, very interesting, especially for binary systems, is that since you know a value of a function of a field by its expansion, on some basis function, it means that in fact you know its value everywhere, not only at the grid points. you know everywhere in space because you have expansion so you have certain on Chibicek polynomials or trigonometrical polynomials and this is very useful when you want to compute suppose you have a field which is defined on the grid centered on one object and you want to compute value at the grid points centered on the other object so usually people are doing when you use finding difference you are doing interpolation have two different grids which do not match and you are interpolated and this is quite

15:00 complicated and especially this is usually very noisy because you don't know how to interpolate smoothly, you have to introduce, this is quite a tricky part of most numerical code once you to interpret. But one of nice features of spectral methods is that you can get the value of a field at any point, so even if it is not the corresponding collocation point, you simply take the spectral the summation, by the coefficient, and you get the exact value, the unique value of the field, and there is no approximation, no truncation error, no noise is introduced by this process. And this is a very nice feature also of the spectrometer. So even though you do have a grid of source, this called location grid, you're not just restricted to the points. Yes, yes, exactly. So in this spirit, it is not really a grid. It is not in the same fashion. It is a representation of a function, but one often says that to compare shortly spectral methods with fine indifference is that with spectral method you are working with functions, I mean you have expansion of basis function, whereas with fine indifference you are working with a set of numbers, values at some points, discrete set of numbers. So you see that when you want to know something between the numbers or between the grid you have to write something and so this is also why these methods are so precise is that you're working really with functions and not with some sets of points. and the set of numbers that you do have the coefficients of the functions what kind of functions do you typically use I mean for the basis functions for the expansion so typically for the radial part so as I told you we are using systematically spherical coordinates so for the radial coordinate R we are using Chebyshev polynomials and we have chosen Chebyshev polynomials maybe for two reasons One is that it is a very nice polynomial to approximate a given function because they have all the zeros allocated between minus 1 and plus 1, so it makes this polynomial oscillate very much, and so they are very flexible in a certain sense to approximate this.

17:30 And the second thing is that Chibicev polynomials, by the definition we can say, are closely linked to trigonometrical functions because you know the cosine of n theta. It is a polynomial in cosine theta, and it is exactly the nf, Chibicev polynomial in cosine theta. And this means that to compute the coefficient of Chibicev expansions, we can reduce this problem as computing coefficient of Fourier series by means of S-link. So it means that practically we can use fast Fourier transform algorithm to compute the Chibicev polynomials. And in numerical splitting, able to use a fast Fourier transform algorithm because it is fast. So this is the two main reasons. It is good to approximate general function, and it is very fast to compute the transformation. For instance, if we used Legendre polynomial, it would be very slow to compute Legendre transform. You have to this n-square operation. And so, this is for the radial part. It's Chebyshek polynomial. For the angular part, theta, the theta part, we are using the cos n theta, with n pseudonymous, trigonometrical polynomials, or sometimes, especially when we are solving a Poisson equation, we're using associated le genre functions, because it's eigenvalue of the Latlesian operator, of course, and so this is for theta. And for the phi part, there is a natural choice, because in phi we are periodic, so we're using simply Fourier there is in five. So this is an obvious choice, I would say. So this is the basis that we're using. And you mentioned that the fast Fourier transformers are so fast that it actually doesn't contribute a lot. Yes, the computational. For instance, what contributes the most expensive part in terms in terms of CPU time, in our course for binary, is when you want to compute the value of a field by taking the whole spectral expansion. If you have a field which is given on another grid, and you want to compute it from the grid of a companion star object, let's say, and you want to compute it at the collocation point of the second object. So, as I told you, you can do this without any truncation error, without any approximation,

20:00 by taking the spectral expansion, but it is a spectral expansion in 3D, in three directions, with Chebyshev, cosine, and so on. And so it is a triple summation, and this is very cost. You cannot reduce it to... You have to do the summation of triple summation. This is the most expensive part of our computation. So how many coefficients do you typically use? Okay, so typically we're using something like 20 or 30 coefficients in R and the number of coefficients is the same as the number of collocation points so you can say we're using 20 or 30 points in R and for high resolution we can increase this to 40 something or even 60 but typically 30 is sufficient and in FITA we are typically using two times less so it means something like between 15 and 15, 20, something like this. And same thing in phi. We're using also 15, 20 in phi. But most of the time you still have good results with only, let's say, typically 17 coefficients in R, only 17 coefficients, and something like 12 coefficients in theta and phi. So it is a very reduced set of points as you compare with finding difference where you're using one and ridden, one and ridden, one and ridden. This is a nice feature of this method. And especially we can reduce, if the object is not too much distorted, if it remains more or less spherical symmetric, we can use very few points, or very few number of coefficients in theta and phi. solution in R, so we can use something like 33 and let's say 8 and 8, so the total number is 8 times 8 times 33, which is quite small, and with this you have a very good description of your star. Whereas when you use Cartesian coordinates, you are obliged to use at least 30 by 30 by 30 at least, even if it is... So this is why we can perform our simulation more or less on workstation. We do not need

22:30 a supercomputer. In fact, we are using SGI Origin 200, which is, the architecture is more or less the same than the Origin 2000, which is the computer that they have in Posidam, for instance. But we have very few, very limited amount of memory. It's only 1.2 gigabyte in each computer, and also limiting number of processors. It is only four processors on the computer. So, if you, so it's, it is not, we can also do it on the PC. For instance, we have a copy of a code with Linux on the PC at all. Of course, it is slow, but it works. I mean, you can do binary in a transfer on the PC because with only you took something like 20 points in R, 12 and 12 in theta phi, and it's okay with, I have 64 megabytes only on my PC, and it works. So, because simply, typical memory for most precise computation for relativistic binary neutron star, it is no more than 250 megabytes. Because simply you have a small number of coefficients. And so are there other groups now interested in using these methods? Yes, I know that there is a group with some thin and KIDR in the United States, they are starting to use spectrometers in numerical relativity. So for the time being, they have only, maybe you are going to interview them, I don't know if you, something. I think, and so they have some preprint with some test problems and also recently they put a preprint with some evolution in spherical symmetry black hole evolution spherical symmetry black hole evolution so it is more as also a test problem but I think 3d things but for the time being they're using only

25:00 for the test problem they represented cartesian types coordinates they didn't seem interested by it well i don't know exactly the reason for this use of cartesian one of the most easiest things that you can do with spectral methods is that you use a cube and Cartesian coordinate x, y, z and you suppose your problem is periodic so that you use Fourier series in each of the three dimensions but this is not well adapted to numerical relativity this is adapted to cosmology let's say you can imagine periodic boundary conditions but periodic boundary conditions are not good of course body for the gravitation field, it is not periodic at all. So you introduce some approximations. I think that one of the main reasons that spectral methods are not so developed in the numerical gravity field is that, but maybe you will investigate this more deeper than I did, because it's only by discussion with colleagues, and it's my who are doing numerical relativity now, they come mostly from the relativistic community. It was people who were doing classical, general relativity before, more or less analytical. And when the computers appear they say, oh, let's now solve the Einstein equation by computer. It seems very attractive and now we can solve 3D things and it would be instead of having simple spherical asymmetric and analytical solution it seems very attractive to a 3D solution. And then they move from this field, from general relativity, to a numerical experiment, we can say, numerical physics. And so they take the most straightforward method which was available, namely finite difference, because you have a partial differential equation, and it is straightforward to write the finite difference part, basically replacing the partial derivative with finite difference. And because they want to have a solution, they were a kind of impatience, they wanted to have a rapid distribution of Einstein's

27:30 equations, and so they take the most, and then, since they began with these methods, they keep, because they develop libraries, you know, and there is a certain inertia, once you start it with something, then you develop, you develop, and I think that if things would to be if numerical relativity would to be developed from scratch let's say now people I think will use instead spectrum methods or finite element methods or other methods but definitely not finite difference because one of the main problems that plug the numerical relativity simulation nowadays seems to be used to link to this to the use of finite difference especially with Cartesian coordinates to describe these spherical objects so the situation would have been different if these people would be coming from hydrodynamics in hydrodynamics spectrum methods are used for more than 30 years or I don't know people if you speak about finding difference to somebody who is doing hydrodynamics they will laugh and say this is just an exercise for students Nobody is using a finite difference, except in numerical relativity. So, because the cost, I mean, the drawback, first glance of spectral methods, it is not so easy to implement, so you have to develop some subroutine, you know, to make this expansion to computer the operator. It is not so straightforward to implement finite difference. But I think that it is really worth to make the effort to implement them. Of course, it takes time. During this time, you don't publish because you are doing only numerics, and you prepare something. So I think that this explains why people who in the past used fine indifference, like Finn, now thinks seriously about moving to Spectrum Method and starts to do things like this because the price to pay is to spend some time to implement some things because it is more complicated

30:00 to implement but once it is implemented it is faster and more precise and more accurate and finding the difference but of course at the beginning to work maybe but this is my opinion so maybe you will have a more precise opinion when you have seen all the groups yeah and i don't know when you were in posdam if you have discussed with about spectrometers with cider because i don't know actually but yeah i knew that he has been always interested by this but apparently he has been he knew that they have so many codes in posdam so many libraries, so many things, that it would have been a big effort now to translate all these things in spectrometrics, so visitate, but I know that Ed is willing to do this at some point, someday, maybe, you know, some, each time I meet him, he speaks about this. Right. So he's interested in it. He's interested in it. You mentioned that it might be the background of people moving into New Mexico, that they were relatives. Yes, yes. That might explain why they just went for finite differencing. Were the backgrounds of people in this group different? Did they come at it from a different angle? I mean, here, yeah, in fact, Silvano Bonazzola was coming, worked in many fields in astrophysics, also in general relativity when he was young, and especially he works also in experimental physics. interesting in the Weber bar type of detecting gravitational waves. So in the 70s, he worked with the experimental team to construct here in Meublon a bar to see whether Weber was right or no. And yes, and then this bar produced no result. So he was interested in gravitational waves and all these things but he knew a lot made a lot of work in high energy astrophysics about x-ray sources gamma and so forth so he has a background let's say of astrophysicist so not of general relativistic if you can say like this and here we are in an observatory so we are in contact with astrophysicist and so this is also why maybe he chose spectral methods because

32:30 he knows hydrodynamics also quite well so he knew some hydrodynamicians who were using spectral methods and as far as Jean Lemarque is concerned, because in the beginning it was both of them, Silvano Bonazola and Jean Lemarque, I think that the first paper they published using spectral methods, it was the gravitational collapse of the stellar core computed by spectral methods it was 86 and Jean-Lenmarc was more mathematically oriented, he did mathematics so he was I think attracted by these methods with all this mathematical background, all these theorems, you know it was expansion with some convergence properties and things like that so it was more appealing for mathematicians to use spectral methods than to use finite difference which are just some I think this is one of the main reasons but Jean-Henri Marc was coming from classical relativity because he studied mathematics and then he made his PhD thesis with Brandon Carter and he studied the geodesics in the Kerr metric and he found some special titrat so it was classical general relativity but I think it is his mathematical background who make him And what was your own background then in relativity before you joined the group here? Sorry? What did you work on yourself before you joined the group? Oh, in fact, I did my PhD already with Silvano Bonazzola and John Lamarck on gravitational collapse of a neutron star into a black hole. And I did it with spectrometers. So, in fact, I started with spectrometers. So, when it was clear from the beginning of my PhD thesis numerical part, the real numerical part will be done with spectral metals. In fact, I studied the neutrino emission from this collapse of neutron star to black hole and studied the propagation of the neutrinos and all this was by means of spectral metals already, so I had no choice and it was I was using spherical coordinates you mentioned that spectrum methods were widely used in other fields years prior to being adopted into the American relativity

35:00 so does that help as far as there are there are textbooks for instance, yes, yes there are very good textbooks written by hydrogen emissions so there is even devoted conferences, each three or two years there is this this is a proceeding of such a conference which is called ICOZAN it means International Conference on Spectral Methods and High Order Methods this one and so it is well spread the methods and there bibliography and also since it it was mathematicians developed some theorems I mean Most of mathematicians doing numerical analysis knows these methods quite well and so on. In the past we discussed with some mathematicians at Paris universities about things and so it was interesting. You can give us some advice. So there's actually, as it were, a community of people interested in speculaments through through which you can hear about new techniques and so on. So it's interesting. We participate in this conference, and it was interesting to discuss with some people doing hydrodynamics, to compare the experience of the problems which are specific to mineral activity, the one which we can share with other people. Are there actually packages that you used that were written by other people or do you just... No, in fact we developed everything from scratch, and when I say we at the beginning it was just in the mid-60s, the only thing that we use, the only library package that we use is for fast Fourier transform because there are many of them of course it's not worth for us to write a new one which would have been worse than the previous one and so fast Fourier transform and also linear algebra standard matrix operations or solving a linear equation with a bounded matrix we are using standard package for this because it is

37:30 all the spectral part part of the differential operators and all the things we developed here in Modon because in fact one of the reasons may be that the use of spherical coordinates this is maybe quite specific to astrophysics because we're dealing with stars if you discuss with hydrogen emission we're dealing with complicated domains, tubes so they're using adapted coordinates or Cartesian types coordinates it is quite seldom to meet people using spherical coordinates even people doing oceanography they are using the boundary condition you have the ground and the ocean so they have to describe this so they are using complicated patches so it's great maybe this is one of the reasons why they have to develop from scratch Because we wanted to use spherical coordinates from the very beginning. You wanted to take advantage of those particularly? Yeah, yeah, yeah. So one of the things I did gather from visiting Potsdam and the Xidal group there is that part of the idea of this European network is to allow the work of different groups using the Cactus code. Is that... Ah, yes. Well, what we did last year is that we provide the Potsdam group with the initial condition for binary neutron star coalescence. I mean, we computed with spectra methods, we computed quasi-equilibrium orbits of binary neutron stars, a dynamical code to compute the full coalescence part. But it is not what they call a cactus thread. It simply will provide the numerical solution. They are using a cubic grid, and since, as I told you before, we are able to compute at any point of space, of course, the cubic grid do not coincide with our spherical collocation

40:00 but with the spectral expansion we compute exactly at each point of the cubic grid we provide to us and we send the result, the numerical value of each metric field of each matter field at this cubic grid. So we did not plug something into cactus directly we provided initial conditions for cactus code and I don't see, and I think we will do this also inside the network, because I don't see any, well, maybe I have to discuss more in detail with Ed about this, but I don't see what it means exactly to make this thread of cactus with a spectropart, how does, well, what is clear for me, I mean from scientific point of view is to provide initial condition, but for this, for our code to be a part of a cactus, I don't see, because it is completely different coordinates, so, well, maybe it's not fair. That's clear, because, yes, also the aim is to do physics with this, to produce astrophysically relevant result so it is not to make a big code as big as possible with so many paths we have a difference with other group is that we have we are quite a small team we have a very strong interaction among us we have to be clear to tell you so now unfortunately It's gone. But we have a new permanent researcher with us, his name is Jerome Novak. He did his PhD in the past with us, and now he gets a permanent position with us. and so with Silvano Benazola and I as a permanent staff and we have two students and two postdocs including one European postdoc so it is quite a small team and so we know exactly what the others are doing we are collaborating

42:30 writing subroutines so I would say that each member of this team has more or less a global view of all the library of the letters, of the codes the situation is different with Cactus where there are many persons which are contributing to many parts and it is very big it is a very big thing so I prefer to work in a small team we see more or less the whole project with a global view. Yes, yes, yes, yes, yes, yes, we have the main methods and, of course, we are still trying to use modern methods for developing calls. I mean, we are documenting our calls, we are documenting in HTML language to, because it's easier for us, especially when a new student arrives, you can make the documentation. We are using version control numbers with we're using a modern debugging things, a code analyzer, we're using modern tools, but we're a small team. You mentioned that spectral methods are particularly effective at approximating smooth functions. And so, but presumably, especially when you're trying to start, you have occasionally you have shocks and things like that. Ah, yes. So for a case of shocks or discontinuities, yes, for this, spectral methods are not good at all if the discontinuity is located inside the computational domain. multi-domain method. We can adapt, and we can adapt the domain by coordinate transformation as I told you. We can allow for discontinuity if it is more or less the topology of a sphere. And we did this, in fact, last year by computing rotating strange stars. Strange stars are a kind of neutron star, if you want. It is compact star, but made of quarks instead of

45:00 nucleons, and with a strange quark, hence the name strange star, and they are different from the trans-star because they have a strong discontinuity in the density of the surface of a star. And so would not have been able to compute with a monodomain the spectrum method, it would have been very bad. But we use our multi-domain and we locate the surface of this stars, the boundary between two domains, so that each field is perfectly smooth inside one domain. And so here you recover spectral accuracy mainly that the precision decays exponentially with the number of coefficients and not as a power. In fact if you have a discontinuity in the derivative it decays as the inverse power of the order of this derivative. So it still converges, I mean Except for a discontinuity, a real discontinuity in a quantity, it is bad. But if it is a discontinuity only in the derivative, you have a first-order convergence, a second-order if it is in secondary breeding and so forth and so on. Right. So, but for shocks properly, I mean, if you want to study the coalescence of two neutron stars, here you will have strong shocks, and with a complicated shape. so for this I think definitely spectrum methods are not the good way to study this so for this we have inside the European network, we have a collaboration with the Valencia group, I don't know if you have been already in Valencia University there is Jose Maria Ibanez who is the leader of this group in fact Jérôme Novak who has been just appointed as a permanent researcher He was, last year, a postdoc in Jose Maria Ebenezes' group, and then he started to develop a hybrid code using both Spectra methods and the kind of high-accuracy finding difference, it is called the Riemann Solveur Methods for hyperdynamics, which are very good to resolve shocks. In fact, it is the best method in the world for solving shocks, and it is these methods.

47:30 They are implemented in the Cactus Code for the hypodynamics part by Ibanez's team. Some of the Ibanez students have been to Postdam to implement this into Cactus, namely Tony Font. Now he's in Gershin. He's no longer in Postdam, but he's in Gershin. and so the idea is what I mean by hybrid code the idea is to take advantage of each method in its own field and it means that we will solve the gravitational field equation, the Einstein equation via spectral methods because you have no discontinuity or very simple discontinuity maybe in some derivative or so on and then you can take all the advantage of spectral methods for solving elliptic equations We solve the hydro, so the shocks, via the Riemann solver methods of Ibanez. And as I told you, we can go from one grid to the other because we have two different grids. There is a spectral grid, the collocation grid, and there is a grid for the hydro. And by means of a spectral expansion, we can compute and switch. Of course, it is quite time-consuming, but it does not introduce an error. And this project has already given birth to a code, a spherical asymmetric code for studying gravitational collapse with shock waves, spherical shock waves, and black hole formation in the presence of shock waves. of Inves, and it has been published this year in Astrophysical Journal, so it was post-doctoral work of Jérôme Novak with Ibanez, and now we plan to extend this to 3D codes for really studying the coalescence of neutron star So I think it is a good strategy not to try to use spectral metals when you have complicated shocks for, as I told you if you have a discontinuity which is more or less spherical, you can, by a multi-domain spectral, you can do it, and we did it It works well, but for a complicated shock, I think you should use these Riemann solver methods, also called Godunov-type methods, because they are really good to describe these discontinuities. So this is what we plan to do. And in any case, these type of shocks would only emerge in the actual merger?

50:00 Yes, only in the merger. up to the merger you can still use spectra methods to do the dynamics since everything is smooth it's ok also I think for binary black hole problem spectra method would be good because you have no shock of the equivalent of shockwave all the metric remains smooth everywhere since you are not treating the central singularity it is outside the computational domain usually thanks to the choice of coordinates So in this case, a spectrometer would be fine all the way along, up to the formation of a single final black hole, I think. There's no reason why you cannot use spectrometer as well. But up to now, as a group, you've concentrated on neutron stars rather than looking at the black holes? Well, in the past, we looked mostly neutron stars, sometimes with black holes in our computation, but as final result of some evolution of some collapse so it is asymptotic black holes if you want and we made only some test computation on the KER metric for instance we recovered numerically the KER metrics because we were interested in computing the structure of a self-gravitating torus around the black hole to see You know, this is relevant to central nuclei of certain galaxies, to which are active nuclear nuclei. And this project, in fact, we didn't follow it. We produced only a test code for computing axisymmetric black hole configurations. and when we put no matter so it is vacuum outside we simply check that we recover the Kerr metric with a very high precision I mean we recover the Kerr metric for A over M equal 0.99 so it's quite a rapidly rotating Kerr, we recover it up to, with a relative precision of 10 to minus 11 with only 33, as I told you So it was a good check of the numerical code, but then we did not go further in this direction.

52:30 But maybe in the future, yes, we will plan to do some black holes, spectrometers, because everything is smooth in the black holes, and since you have this, I think it is worth to do it we also have spherical coordinates because of the topology of the horizon, so maybe we'll move in this direction. You mentioned that you make use of multiple domains. Yes. And in particular, one example being that you have a special domain, a compactified domain. Yes, yes. Very exterior. Yes, yes. And I know that people try to use, I guess, somewhat effectively a similar idea in grids where they have Cauchy characteristic matching. Presumably you also have to match between the domains, and is that much easier? Well, this compactification, it is good for elliptic operator, not hyperbolic ones. I mean, for only, basically, when you numerically solve for the Einstein equation, you have some equation, it depends strongly on the choice of coordinates, of course, that you are using, but basically, very often, you have a set of elliptic equations, like, say, the equation for the lapse function in maximal slicing, for instance, typically, the appellation of the lapse function equals something. so very often the Hamiltonian constraint equation is an elliptic equation for the conformal factor of the free matrix so for this elliptic equation it is very good to introduce this compactified domain because you can really solve you need to put some binary condition since it is an elliptic equation and it is very nice to be able to put it to infinity because you put a flat space time value For the hyperbolic part of the Einstein equation, the part which contains gravitational waves in particular, of course, here you have to divide something more adapt to this, and you mentioned the Cauchy characteristic matching, so you have to match some 2 plus 2 formalism, but we haven't tried yet to have spectra methods to do such matching, 2 plus 2.

55:00 but there must between the different domains that you already have there must presumably be some sort of matching that you have to do yes in each domain so in fact we can say that all these domains are on the same hyper-surface spatial hyper-surface and so all the matching is that across the domain, so it's very easy to implement the spectrometers since when you know, really we deal with functions as I could do, so it's easy to say that a function should be smooth because simply you can look at its derivative and very easy, it's very easy and very fast to compute a derivative with spectrometers, it's a very simple operation and you have a very good accuracy in the derivative. so it's quite easy to this condition it is not a problem at all right so it's quite good for imagining yeah yes yes yes the so well the context in which I first became familiar with the work of group here was when you produced some work a year or two ago now on binary neutrons in which you were able to amongst other things replicate or make as it were contribution to a debate have been going on Yes, well, the Wilson results, yes as I recall you found no No significant compression, yes In fact, there is a slight compression as we saw in our curves and since then we have computed more relativistic configurations with a higher compactification ratio, NISTAR And this compression, I mean increase of central density along the sequence when the two star approach, this increase is slightly more important with more relativistic stars, but it is something like at most 1% or something like this, so it is not very significant, I would say.

57:30 especially we introduce in all this computation we introduce some approximation it is that we take the spatial free metric to be conformally flat and this is an approximation for binary system because it should not be exactly conformally flat but this is I think quite a good approximation but the error introduced by this approximation is also of this order 1 or 2% I don't know it's difficult to say because we have no exact solution to compare with so it's difficult to gauge to gauge this approximation but let's say so if you see an increase of only 1% maybe this is due only to this approximation, maybe it is not physical so you cannot conclude anyway it is not 10 or 20% or more the first result of Wilson and Matthews but now we know These results were due to some error in some analytical formula. In fact, you know maybe that, I don't know if you follow this story, it was Flanagan who did a paper about this. And in fact, I noticed it is an error in the Lorentz factor, something like this. It is a kind of basic error. And so I noticed already this error when I read the paper by Wilson and collaborators. it was a misprint error because it was such an elementary formula that I thought, and I know that Wilson is very familiar with relativistic hyperdynamics, so I thought it was simply a misprint and I didn't imagine that it could be the source. In fact, it was really the source of the problem. So subsequently, when they, as I recall, when they incorporated Flanagan's correction, there was a problem, they reformed, they rewrote their coder, and then they said they still... Yes, they still find something which is at a lower level, but they still find this effect. But as I told you, I think this effect is within the error bar of the conformity flat approximation. So I know that they conclude, they try to keep the same conclusion,

1:00:00 but I think it is not very, very, very fair. One should be more careful about this effect. the conformally fat error, so, and anyway, it is very small effect, so, and to tell you the truth, since you are making some historical account about this thing, at the beginning when we started this project, we found that the criticism against Wilson and Mathews had been very strong and more or less unjustified. I mean, most of the criticisms were not relevant to the computation because the computation was the first relativistic, fully relativistic apart from this conformity flat approximation, but first fully relativistic computation of such a thing. And most people object against them by saying well, in Newton's case, it does not appear or if a point particular, it does not appear. so most of the criticism were not relevant in our opinion you cannot say we don't have the experience of this of course it's new results when we started this project we had no a priori about Wilson results, we said maybe Wilson is right, maybe he's wrong but maybe he's right so let's have a look in fact I would even like to confirm who's another because i found that the criticism against wilson's team most of them may be quite unfair and quite strong if you look yes and also i remember in some conference some people were more or less laughing against them and it was not very i didn't like this And, well, but unfortunately our results show that Wilson was wrong and so it was better. So you had specifically set out to try to replicate... No, it is not a replication of the results. The results came from dynamical simulation, more or less dynamical, because it is dynamical for the hydro part, it is full hydro. but for the gravitational part

1:02:30 there is no real dynamics they solve more or less only the constraint equation so it is this kind of thing in our computation it is not dynamical but we look at irrotational binary states so it is equilibrium state on equilibrium orbit and we look at states which more or less correspond to the n-state of computation it appears in the computation that the stars more or less keep the vorticity as they approach each other and now we understand this quite well on physical ground with estimates of the viscosities in neutron star matter matter we know that more or less it should be at the end it should be i mean if it is irrotational in the beginning i mean you can stars with respect to the final orbital velocity. I mean, so if you start with a spin of a period of rotation of one second, typical pulsar, typical neutron star, it is negligible with respect to the kilohertz frequency at the end. So it means that physically when we are clause, the fluid should be more or less irrotational, as it were in the beginning, and this is the difficult part to compute, because previous to a computation, there were computation by an American team led by Baumgarter, Shapiro, and Tokolsky, which were co-rotating binary in a trans-star in equilibrium. So this is more easier to compute, because in co-rotating, if you put yourself in the co-rotating frame, the situation is static. You have no hydrodynamics so you have only to solve the Einstein equation. But in the irrotational case, if you put yourself it is natural of course to put yourself in a frame which is orbiting with the stars but in this frame you have a velocity field and you have to compute this velocity field so this is why it has not been done before because it is more complicated. You have to introduce some hydrodynamics and so on. So this is we did this problem with spectrometers, because it is so good to put a good boundary condition on the velocity field and so on.

1:05:00 And so our computation our equilibrium solution and we simply make sequence by identifying I mean two equilibrium states by stating that they should have the same the quantity which is conserved during the evolution is the barium number, obviously. And so this is the way we make sequence. We compute various equilibrium orbits, more and more tight, and we demand that the barium number should be constant along these orbits. And in doing so, we can see how the central density evolves or not. So it is not a replication of, it is a different method, different is an approximation of equilibrium orbits. Whereas Wilson, it was really a time evolution for the hydro, and so it, well, this is right. Yeah, no, I see what you mean. Although, didn't they, and here, not much of an expert, didn't they conform a flatness assumption that Wilson-Matthews made mean that, in some sense, Yes, for the gravitational field, we can say, because this conformally flat approximation forbids the existence of gravitational wave, so in a certain sense, yes, it kills the dynamics of the gravitational field, but it does not kill the dynamics of the matter field, the hydro. So they were doing really hydrodynamics, but it is not, in fact, the hydro, it is not so important in this stage. the evolution up to the coalescence. It is more or less a succession of quasi-equilibrium states. So this approximation is very good to use this. So in your approach, you have an equilibrium state not only in the gravitational field, but also in the matter field. It is equilibrium, but not static. There is a velocity field which corresponds to irrotational flow, but it is Actually, you define it as an irritational flow because the spin is...

1:07:30 Yes, the motivation is that if you look at the various spins involved in this field, so you have the initial spin of each neutron star, and you have the final orbital frequency, And for typical values of neutron star spins, I mean not millisecond. Of course, if you take millisecond pulsar, it is not irrotational. But if you take a pulsar of period one second, it is negligible with respect to the final frequency, which is in the millisecond range. so you can approximate this one second to zero zero frequency and so if it is zero it means that initially your star is irrotational and since you have non-dissipative fluid, the viscosity is highly negligible in all this stuff because the time scale evolution is much too fast gravitational wave driven time scale is too rapid for viscosity to synchronize the stars remain irrotational in the way around. So they'll still be not spinning? They should be not spinning in the end. This is a very good approximation. Because the time scale required to synchronize the stars to make them spin where the orbital frequency is matched to learn by many orders of magnitude something like 7 or 8 orders of magnitude So it's a very good approximation to make them irrotational. But then, as you say, if you do have them be irrotational, then in a co-rotating frame, then they actually are spinning. Yes, they are spinning. They are spinning in the opposite direction, in the direction opposite to the orbital velocity, so they are counter-rotating, so we say. It is counter-rotating the transstand instead of co-rotating. It is counter-rotating. But since then, two other teams have obtained such, since our first result that you mentioned, it was published something like one year ago. Yes, it was in the beginning of 1999. But since then, two other teams have developed similar approximations, irrotational equilibrium

1:10:00 binaries. It is Wilson and Matthew's team, Maronetti itself, so they build a different code, different from the dynamic code, so they will find the difference. and there have been the Japanese team by Iriguchi and Uryu and it is going to be published in the physical review this year and they obtained results which are similar to ours so no big surprise so as you say it's not a replication of the semantic method but were you inspired to produce this particular version of the code Well, we were thinking at some point, because in the past, we studied the ice-related and before this project, we were studying because we have always been interested in gravitational waves, probably emission, in connection with Virgo project or LIGO project, of course. We have strong link, personal link, with people in Varigotin, especially Alain and so we were studying by which process isolated single neutron star could emit periodic gravitational waves and we studied various process deformation by the magnetic field so we compute with spectrometers we solve also the maxwell equation so the Laplace forces and to see whether Virgo could detect such neutron star or not. We also studied the symmetric breaking of rapidly rotating neutron star some instabilities which developed in rapidly rotating neutron star to see if in some x-ray binary system we can have a good gravitational wave limit. So we were at that stage when Wilson Matthew's results late in 95 or 96 but we were thinking about doing binary neutron star in the future because of course it is obviously interesting for gravitational waves it was a strong motivation for us

1:12:30 yes to investigate this controversy this result but at some point we would have been in this field of binary system because we did We sanitized the TronStar, we finished with it, we did some kind of symmetry breaking through the instability, and also it was before the appearance of the R-Mod, you know, this R-Mod and stuff and so on. So we were not aware of this R-Mod, so we considered at that time that it was more or less finished. Nothing is really finished, you know, in science, but we wanted to move to completely new things. around this time we had started anyway we would have started anyway this binary in the TronStar maybe a year after let's say Wilson and the Mathews it was simply for us to urge them to do it now a year after we would have started binary in the TronStar have you been inspired then by this or mode to go back oh yes Exactly. So one PhD student just began his PhD this year. He's working on instabilities in rapidly rotating the trans star. This is inspired by this armor. Because we have all these codes and we will look at this. but it's just beginning it's just starting so is it your plan to go on and do dynamical simulations yes we have various directions and one direction is black holes of course because obviously it's interesting for gravitational emission and the other direction is to have dynamics in binary in the trans stars and for this we are as I told you collaborating with José María Ibanez in this European framework in Balencià to put the hydro not with spectrum method but with this Godot-North type method and we are currently working on this so this is our two main

1:15:00 and in the dynamics we want of course to leave this conformally flat approximation because for binaries all our computation from now have this conformally flat free matrix now we want to remove this we have already done the analytical part written on the equation so in this case it is the full 3D Einstein equation It's quite complicated, but to work with this, we have developed something also which I think has not been done to this extent in the other groups. is that I would say now we have two main differences of a numerical relativity group one is that we are using spectrum methods instead of finding difference and the second is related to the computational language we abandoned Fortran which is used by most groups two years almost three years ago three years ago in the beginning of 87 We move, we migrate to C++ programming language. It means that it is completely different for us because it is object-oriented language. So the philosophy of programming is completely different. And we can manipulate more abstract objects than with Fortran. And especially we manipulate tensor fields. we have a kind of language now which is based on C++ and which looks like Mathematica if you want you have addition of you have tensile operation you do no longer have loops loops are not the number of points which are typical of Fortran codes and even stars one star is considered as an object so you have a star with various tensor fields so you have a kind of object which is a binary system it is composed of two stars which are C++ objects by themselves and so you can for instance write read a star on a file it's simply constructing with a constructor C++ from a file and you have all this object oriented it is object oriented way long and for us it was really a revolution to use this object oriented

1:17:30 language because now the development of our code is much more faster and much more secure than in the past in fact we simply regret not to have abandoned Fortran sooner because you know at Fortran we spend our time checking all the loops the boundary because you have to be careful about the size of the array not to go to overflow and so you're all tricky and dirty things can arise and we spend a lot of time just checking finding some errors some stupid numerical errors some mistakes and with C++ it became much more secure because we do not have loop any longer we do not care about dimension because it is implemented since we manipulate now quite abstract things we have at the beginning course, we take care about some loops in the very deep subroutines, because at the end, of course, you make some addition of some arrays of numbers somewhere in the computer, of course, it still exists, of course, but since you have defined, for instance, suppose you want to define the addition of two tensor fields, two scalar fields, so you do it properly once, you check everything, and once it is done, you have an addition operator, and it works for your life and you simply add a plus so you have plus A plus B we write like this A plus B if A is a vector field the plus will check that B is also a vector field, if it is a rank 2 tensor it will check that B is a rank 2 tensor unless you will have an error message so it is much more secure to type A plus B so for us it was very very good to move to C++ we don't At the time, of course, at the beginning of 1997, it took a lot of time for us to do only numerics, I mean, to re-implement everything, to rewrite all of it. One of the advantages of C++ is that we could call from C++ all the Fortran subroutines. So we do not have to make everything from scratch. We could grow progressively. Now, most all of the Fortran part has been removed, so we have only C++.

1:20:00 except from subroutines which comes from a package like fast Fourier transform typically it works, there is no need of course to rewrite in C++ because we have good Fortran, it has been tested by so many people in the world I mean it's not our job to write this in C++, we want to do astrophysics but all the new things we develop in C++ systematically and also from my experience when a new student comes it's much more easier for him to manipulate spectrometers on a with C++ because he can concentrate on the physical equation on the physical problem he's not sparing most of his time by checking the he can go very fast into the physical it's nice in this respect also We have experienced this with students. Also with foreign collaborators who come here just for one month to work with us, it's much more rapid for them to also to do this. They can, for instance, use our code and introduce some modifications. They want to print out something, to have a look in some particular quantity. it's much more easier for them to do this than to go inside the Fortran code to localize where exactly you are doing this it was really complicated and also one of the motivations to move to this object oriented language appeared clearly when we started this binary stuff because in the past we were a Fortran code we had single so a single set of coordinates. But then when we move to a binary star, we begin to have two coordinates. Entrez! You were saying that... Yes, yes. When we start to move to this binary star, we realize that we have this multi-domain method, so we have multi-domain, and we have two sets of coordinates, which are centered on one star so we have some fields like density field which is defined on one star another star. It would have become very complicated to do this in Fortran I think. You would have some

1:22:30 arrays with high number of dimensions because you have dimension for the index tensors. Basically we have something like two tensors with two indices metric tensors and something. And then you have index for spatial position r theta phi three index, three indices you have indices for the domain and then indices for the stars so you can count it seven indices so it becomes quite messy with Fortran so this is also one of the motivations since we all things become complex with this binary a higher level language like C++ was definitely more adapted to this than quite a basic language such as Fortran. You were saying that when visitors come, it also sort of easier for them. Does that even hold true if they are not previously familiar with C++? Yeah, we have visitors which were familiar with C because the syntax is familiar. Of course, at the beginning I had to introduce them to C++, especially to object-oriented programming. So, but since then, since they knew quite a bit C++, now they can arrive and they can go into a new code without any difficulty. Of course, yes, you have to know a minimum of C++, yes, to understand. But since you know the syntax and you know the philosophy of the language a little bit, what is object-oriented programming, then if someone gives you a code that you don't know you don't know this code, you can go and understand the code much more rapidly than Fortran for sure because A plus B you see more readable you mentioned you were quite close Yes, in fact, formally, even Jean-Lenmarc was a member of the Virgo team, formally. He attended to the Virgo project. Also, he was participating in software development for Virgo. He was participating in the project for a numerical simulation of a wool experiment,

1:25:00 for studying the noise, various numerical noise in the experiment. So he was devoting, let's say, 20% or something like this of his time to this, not to numerical relativity, but to this. And also we have contact and discussion, regular discussion with Alain Brier, which is the French leader of Virgo team. So we see him from time to time, and we are discussing about the gravitational wave sources. we also made some paper with Silvano and Alberto Giazzotto which is the Italian leader of Virgo about the detectability of neutron stars in the galaxies and we look in this so we have closely with Virgo but formally Silvano, Bonazzola and I are not members of the Virgo team Jean and Marc were members but we are not members team, I think the philosophy is quite different than in LIGO. In LIGO there are astrophysicists, there are general activists, which are a member of LIGO. You know this better than I. But in Virgo people consider that it is an experiment and it is mostly done by particle physicists. And at least for the French particle physicists, they make a clear difference between theory and experiment. And usually in In people working with particle accelerators, they are only, they consider themselves experimentalists, and if you are involved in this, you are ready to do something on the experiment. So this is why Jean Lamarck, when he was a member of Virgo, he was really doing something on Virgo. He was making some software for simulating the whole experiment. You cannot be a member of the Virgo team if you are not working directly for the Virgo experiment. You cannot say, well, I'm an astrophysicist interested in gravitational wave sources. you have to so at a certain point people from Virgo asked me and Silvano Benazola do you want to be a member of Virgo and they asked also to Luc Blanchet has been asked but they told us if you want, if you say yes you will

1:27:30 spend at least 10% of your time to do something on Virgo and since we have so many projects and so on, I did not want at that time to get involved in something completely different, so I said no. But Jean and Marc said yes. So this is our link with Virgo. So we are regularly invited in Virgo conferences, for instance, when there are conferences, we are regularly invited, so we discuss with them, and we meet from time to time, and so this And do they suggest to you, you know, this is a type of source that we think would be particularly interesting? No, in fact, it is, maybe I would say it is the reverse, because they know very little about astrophysics, about neutron stars, about black holes, because they came from nuclear, nuclear physics, most of them, or optics, like Alain Brie comes from laser, laser optics. So you were saying that usually it's the case that you, that because of the background of the Virgo people in particle physics... Yes, yes, so usually they are interested, for instance, they do not read the aproposical literature. So even if it is not a product from our group, but when we read something interesting about the R-Mod we told them this is a new source this might be so this is in fact I know that they would like to have astrophysicists working 100% of time for them it would be very interesting if something can make a compilation of all possible sources and be aware of the literature about sources that would be very interesting in this. But we can do not play this whole fully because we are involved in this numerical relativity stuff, so we do this only just by discussion. And do they also have interest in having someone, having astrophysicists who are interested in

1:30:00 Yes. In fact, this year in the CNRS, which is our agency for permanent position in research in France, they opened a position specifically devoted to data analysis in Virgo, permanent position. So, we don't know the result of the concourse, because it is a concourse. So, many people apply. So, there will be somebody working specifically for data analysis. So, it's a special appointment just for that purpose? Yes, just for this purpose. And that person is, say, envisaged to be someone from an astrophysics background? No, he's not supposed. He's supposed preferably to have a good background in data analysis, so maybe more or less from a mathematical point of view or a computational science point of view, something like this. Maybe somebody who has worked before on a nuclear physics experiment So, something was some background with that analysis. So, how long do you think it would take to, say, do a fully dynamical neutron star simulation or a fully dynamical simulation of a binary neutron star that would produce waveforms? Yeah, it's difficult to say, as always. I think it's a matter of years, a few years, but to have reliable gravitational forms, because one has two extra room, quite a pair of arms. So, but it would be, of course, very interesting to have such in order to confront with the observation, to have some knowledge on the equation of state, you know, if you observe this way from, you can tell the equation of state is such and such, and so it is very interesting for the physics of the interior of neutron star, the atomic physics. and that of well I cannot tell you maybe you know that Shibata recently made very interesting progress in this direction

1:32:30 he already obtained the first fully 3D dynamical with gravitational full Einstein time evolution for binary neutron stars it is very recent result and I think really the person which is the more advanced in this direction is Shidata. And of course it is first result, so it is still coarse resolution, the gravitational waves are extracted quite close, so it is not high label for I think for Virgo or LIGO, but it is first important step in this 3D GR. And it's quite impressive how much of it is. Yeah, it is. Especially since he's not alone, but he's working with only a few persons, Uruyu, maybe a little bit Shapiro, and if you compare this with some big teams, it's really a big achievement. I was very impressed by you. He will visit us next June, so I'm quite happy to be able to discuss directly with him about this binary neutron star evolution. He will give a seminar and he will stay for 10 days, so we will have a discussion about this next month. Next June, yes. Yeah, he's certainly very impressive. What would you say are the main difficulties that would have to be overcome in your own code? Well, for binary neutron stars, I'm not speaking about black code, but neutron stars, in principle, it's only a matter of work, I mean, to implement all the Einstein equations. and also to think something yes, maybe the most difficult part is the boundary condition for the gravitational field for the gravitational wave part of the field is to divide some some approximation for, you know, outgoing waves because you have all this problem this is independent of the numerical methods if you find a difference, a spectral

1:35:00 you will get incoming waves. So in the past, Bonazzola and Mark have already played wave equations with spectrometers. So we have some experience about solving wave equations with So we plan to take this experience into account for these binaries, but I think it's maybe... Yes, I would say this is the most difficult part is the boundary conditions for gravitational waves. And is that most difficult because of, say, correctly matching it to...? As I said to you, it is independent of the numerical methods, so either you use a characteristic code outside or you use some well-chosen outgoing web solution. We are also thinking about working with this with Luc Blanchet because Luc has a good experience of course of post-Newtonian expansion and so maybe we can match with some analytical Luke calculation. We are currently discussing this with him. So maybe this is one good solution. But we don't know yet. In fact, this is in progress. But that's probably the main conceptual thing that you would have to overcome. Yes, yes, yes. Other than that, it seems... Yes. Otherwise, I would, yes. If you look at the full Friedenstein equation, there is nothing special. I mean the non-linearities are rather weak if you compare with the non-linearities of hydrodynamics so it's only quadratic in the derivative so it's no problem with spectral methods no, it's just a method to implement them to check everything so I think the most difficult part is the boundary condition and computing resources probably wouldn't be a problem either? some estimations since i thought already we have quite limited computers i think it will require something like one gigabytes so this is we can allow for this right now and it seems

1:37:30 this is not a real problem i would say because of course if it is not sufficient we can ask CNRS for new computers. From time to time we are making a demand. We need this type of computer to do this project. And of course we should motivate our demand. And most of the time we have the grant and we can buy new computers. so it's not a problem to if we say ready to make this project we have all the equations, we have all the methods we know how to put the boundary condition we know everything, we are only missing a new computer CNRS will give us a new computer no problem except if we demand of course a huge computer gigabytes of course it's a matter of cost but if it is something like In France, is there an equivalent to things like the NCSA in the U.S., or places where they are? Yes, voilà, voilà, supercomputing center, so you can submit jobs to the CS, it exists. But our philosophy here is always, we are in perpetual development, we are in developing culture. So we want to check, to test, to make some more runs. So for us, it's why we don't want to submit to get those at one week after. It's not our way of... So, because one of the advantages is that, as I do, we have quite small computers, but it is entirely devoted to our team. We are the only user of this computer. So we have Android percent of time on it. So even if it is only four processors, it is not 64 processors, it's only four processors. In fact, we have two computers, so we have eight processors, but these eight processors are for our own usage, use. So, well, it's not too bad a situation. Yeah, sure. In the past, Jean-Lenmar and Silvia Nobunazawa submitted some jobs on Cray, in a supercomputing center in France, because at that time the computer that we have here in Moderna was not sufficient for. So it was the first, they computed the first 3D gravitational collapse, in Newtonian theory, of a stellar core.

1:40:00 So it was 3D, and at that time, even Newtonian 3D was quite demanding in terms of CPU resources. So it was not able to do it on a local computer, and they did it in a super computing system. Excuse-moi. So, can we entree? Okay. So they did use it in a supercomputer in the past, but... In the past, but now our local computer is sufficient for our purpose. Because, as you say, you're sort of into that... Yeah, so we want to make... And we want to have results immediately, because we want to see if we should change something. And as I told you, this is one of the advantages of spectrum metals, we need less computational resources because we need less memory, because we have less points, so we can do it on small computers, not super computers, but small computers. but so although you do intend to look at dynamical evolution of binary neutron stars or maybe binary you're also still interested in single neutron stars yes yes one of our students yes one of our students as i told you is looking in this because in this case for these 3d instabilities you have no shock waves, so you can do also the hydro by spectra method. So you don't have event or everything is smooth. I think it is feasible as a Ph.D. for this type of thing. Do you, for the different problems that you tackle, such as this individual neutron star and the binary neutron star, do you maintain entirely separate codes? Yeah, no, yes, it is a separate code, but it is based on the same, we use the same abstract object, the C++ tonsorial fields, and so we have this C++ library, which we call the And so we were using Loren for the main part, and from Loren we built entirely different code, because of course if one star is different than two stars, but we used the same abstract object stars in both codes, the same C++ class.

1:42:30 So, I guess maybe we can leave it there. Okay. Thank you very much. That was really very interesting. I don't know if my English is quite understood.