Modellansatz - English episodes only

Sandra May works at the Seminar for Applied Mathematics at ETH Zurich and visited Karlsruhe for a talk at the CRC Wave phenomena. Her research is in numerical analysis, more specifically in numerical methods for solving PDEs. The focus is on hyperbolic PDEs and systems of conservation laws. She is both interested in theoretical aspects (such as proving stability of a certain method) and practical aspects (such as working on high-performance implementations of algorithms). Sandra May graduated with a PhD in Mathematics from the Courant Institute of Mathematical Sciences (part of New York University) under the supervision of Marsha Berger. She likes to look back on the multicultural working and learning experience there.

We talked about the numerical treatment of complex geometries. The main problem is that it is difficult to automatically generate grids for computations on the computer if the shape of the boundary is complex. Examples for such problems are the simulation of airflow around airplanes, trucks or racing cars. Typically, the approach for these flow simulations is to put the object in the middle of the grid. Appropriate far-field boundary conditions take care of the right setting of the finite computational domain on the outer boundary (which is cut from an infinite model). Typically in such simulations one is mainly interested in quantities close to the boundary of the object.

Instead of using an unstructured or body-fitted grid, Sandra May is using a Cartesian embedded boundary approach for the grid generation: the object with complex geometry is cut out of a Cartesian background grid, resulting in so called cut cells where the grid intersects the object and Cartesian cells otherwise. This approach is fairly straightforward and fully automatic, even for very complex geometries. The price to pay comes in shape of the cut cells which need special treatment. One particular challenge is that the cut cells can become arbitrarily small since a priori their size is not bounded from below. Trying to eliminate cut cells that are too small leads to additional problems which conflict with the goal of a fully automatic grid generation in 3d, which is why Sandra May keeps these potentially very small cells and develops specific strategies instead.

The biggest challenge caused by the small cut cells is the small cell problem: easy to implement (and therefore standard) explicit time stepping schemes are only stable if a CFL condition is satisfied; this condition essentially couples the time step length to the spatial size of the cell. Therefore, for the very small cut cells one would need to choose tiny time steps, which is computationally not feasible. Instead, one would like to choose a time step appropriate for the Cartesian cells and use this same time step on cut cells as well.

Sandra May and her co-workers have developed a mixed explicit implicit scheme for this purpose: to guarantee stability on cut cells, an implicit time stepping method is used on cut cells. This idea is similar to the approach of using implicit time stepping schemes for solving stiff systems of ODEs. As implicit methods are computationally more expensive than explicit methods, the implicit scheme is only used where needed (namely on cut cells and their direct neighbors). In the remaining part of the grid (the vast majority of the grid cells), a standard explicit scheme is used. Of course, when using different schemes on different cells, one needs to think about a suitable way of coupling them.

The mixed explicit implicit scheme has been developed in the context of Finite Volume methods. The coupling has been designed with the goals of mass conservation and stability and is based on using fluxes to couple the explicit and the implicit scheme. This way, mass conservation is guaranteed by construction (no mass is lost). In terms of stability of the scheme, it can be shown that using a second-order explicit scheme coupled to a first-order implicit scheme by flux bounding results in a TVD stable method. Numerical results for coupling a second-order explicit scheme to a second-order implicit scheme show second-order convergence in the L^1 norm and between first- and second-order convergence in the maximum norm along the surface of the object in two and three dimensions.

We also talked about the general issue of handling shocks in numerical simulations properly: in general, solutions to nonlinear hyperbolic systems of conservation laws such as the Euler equations contain shocks and contact discontinuities, which in one dimension express themselves as jumps in the solution. For a second-order finite volume method, typically slopes are reconstructed on each cell. If one reconstructed these slopes using e.g. central difference quotients in one dimension close to shocks, this would result in oscillations and/or unphysical results (like negative density). To avoid this, so called slope limiters are typically used. There are two main ingredients to a good slope limiter (which is applied after an initial polynomial based on interpolation has been generated): first, the algorithm (slope limiter) needs to detect whether the solution in this cell is close to a shock or whether the solution is smooth in the neighborhood of this cell. If the algorithm thinks that the solution is close to a shock, the algorithm reacts and adjusts the reconstruted polynomial appropriately. Otherwise, it sticks with the polynomial based on interpolation. One commonly used way in one dimension to identify whether one is close to a shock or not is to compare the values of a right-sided and a left-sided difference quotient. If they differ too much the solution is (probably) not smooth there. Good reliable limiters are really difficult to find.

• S. May, M. Berger: An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes, Preprint available as SAM report, number 2015-44, 2015.
• S. May, M. Berger: A mixed explicit implicit time stepping scheme for Cartesian embedded boundary meshes, Finite Volumes for Complex Applications VII - Methods and Theoretical Aspects, pp. 393-400, Springer, 2014.
• S. May, M. Berger: Two-dimensional slope limiters for finite volume schemes on non-coordinate-aligned meshes, SIAM J. Sci. Comput. 35 (5) pp. A2163-A2187, 2013.

If we are interested in the propagation of waves around a small region of interest, like e.g. an obstacle inside a very big ("unbounded") domain, one way to bring such problems to the computer and solve them numerically is to cut that unbounded domain to a bounded domain. But to have a well-posed problem we have to prescribe boundary conditions on the so-called artificial boundary, which are not inherent in our original problem. This is a classical problem which is not only connected to wave phenomena.

Sonia Fliss is interested in so-called transparent boundary conditions. These are the boundary conditions on the artificial boundaries with just the right properties. There are several classical methods like perfectly matched layers (PML) around the region of interest. They are built to absorb incoming waves (complex stretching of space variable). But unfortunately this does not work for non-homogeneous media.

Traditionally, also boundary integral equations were used to construct transparent boundary conditions. But in general, this is not possible for anisotropic media (or heterogenous media, e.g. having periodic properties).

The main idea in the work of Sonia Fliss is quite simple: She surrounds the region of interest with half spaces (three or more). Then, the solutions in each of these half spaces are determined by Fourier transform (or Floquet waves for periodic media, respectively). The difficulty is that in the overlap of the different half spaces the representations of the solutions have to coincide.

Sonia Fliss proposes a method which ensures that this is true (eventually under certain compatibility conditions). The chosen number of half spaces does not change the method very much. The idea is charmingly simple, but the proof that these solutions exist and have the right properties is more involved. She is still working on making the proofs more easy to understand and apply.

It is a fun fact, that complex media were the starting point for the idea, and only afterwards it became clear that it also works perfectly well for homogeneous (i.e. much less complex) media. One might consider this to be very theoretical result, but they lead to numerical simulations which match our expectations and are quite impressive and impossible without knowing the right transparent boundary conditions.

Sonia Fliss is still very fascinated by the many open theoretical questions. At the moment she is working at Ecole Nationale Supérieure des Techniques avancées (ENSTA) near Paris as Maitre de conférence.

• C. Besse, J. Coatleven, S. Fliss, I. Lacroix-Violet, K. Ramdani: Transparent boundary conditions for locally perturbed infinite hexagonal periodic media, arXiv preprint arXiv:1205.5345, 2012.
• S. Fliss, P. Joly: Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media, Applied Numerical Mathematics 59.9: 2155-2178, 2009.
• L. Bourgeois, S. Fliss: On the identification of defects in a periodic waveguide from far field data, Inverse Problems 30.9: 095004, 2014.

How do populations evolve? This question inspired Alberto Saldaña to his PhD thesis on Partial symmetries of solutions to nonlinear elliptic and parabolic problems in bounded radial domains. He considered an extended Lotka-Volterra models which is describing the dynamics of two species such as wolves in a bounded radial domain:

(u1)t−μ1Δu1=a1(t)u1−b1(t)u12−α1(t)u1u2(u2)t−μ2Δu2=a2(t)u2−b2(t)u22−α2(t)u1u2

For each species, the model contains the diffusion μ of a individual beings, the birth rate a, the saturation rate or concentration b, and the aggressiveness rate α.

Starting from an initial condition, a distribution of u1 and u2 in the regarded domain, above equations with additional constraints for well-posedness will describe the future outcome. In the long run, this could either be co-existence, or extinction of one or both species. In case of co-existence, the question is how they will separate on the assumed radial bounded domain. For this, he adapted a moving plane method.

On a bounded domain, the given boundary conditions are an important aspect for the mathematical model: In this setup, a homogeneous Neumann boundary condition can represent a fence, which no-one, or no wolve, can cross, wereas a homogeneous Dirichlet boundary condition assumes a lethal boundary, such as an electric fence or cliff, which sets the density of living, or surviving, individuals touching the boundary to zero.

The initial conditions, that is the distribution of the wolf species, were quite general but assumed to be nearly reflectional symmetric.

The analytical treatment of the system was less tedious in the case of Neumann boundary conditions due to reflection symmetry at the boundary, similar to the method of image charges in electrostatics. The case of Dirichlet boundary conditions needed more analytical results, such as the Serrin's boundary point lemma. It turned out, that asymtotically in both cases the two species will separate into two symmetric functions. Here, Saldaña introduced a new aspect to this problem: He let the birth rate, saturation rate and agressiveness rate vary in time. This time-dependence modelled seasons, such as wolves behaviour depends on food availability.

The Lotka-Volterra model can also be adapted to a predator-prey setting or a cooperative setting, where the two species live symbiotically. In the latter case, there also is an asymptotical solution, in which the two species do not separate- they stay together.

Alberto Saldaña startet his academic career in Mexico where he found his love for mathematical analysis. He then did his Ph.D. in Frankfurt, and now he is a Post-Doc in the Mathematical Department at the University of Brussels.

• A. Saldaña, T. Weth: On the asymptotic shape of solutions to Neumann problems for non-cooperative parabolic systems, Journal of Dynamics and Differential Equations,Volume 27, Issue 2, pp 307-332, 2015.
• A. Saldaña: Qualitative properties of coexistence and semi-trivial limit profiles of nonautonomous nonlinear parabolic Dirichlet systems, Nonlinear Analysis: Theory, Methods and Applications, 130:31 46, 2016.
• A. Saldaña: Partial symmetries of solutions to nonlinear elliptic and parabolic problems in bounded radial domains, PhD thesis, Johann Wolfgang Goethe-Universität Frankfurt am Main, Germany, 2014.
• A. Saldaña, T. Weth: Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains, Journal of Evolution Equations 12.3: 697-712, 2012.

This episode discusses the Born-Infeld model for Electromagnetodynamics. Here, the standard model are the Maxwell equations coupling the interaction of magnetic and electric field with the help of a system of partial differential equations. This is a well-understood classical system. But in this classical model, one serious drawback is that the action of a point charge (which is represented by a Dirac measure on the right-hand side) leads to an infinite energy in the electric field which physically makes no sense.

On the other hand, it should be possible to study the electric field of point charges since this is how the electric field is created. One solution for this challenge is to slightly change the point of view in a way similar to special relativity theory of Einstein. There, instead of taking the momentum (mv2) as preserved quantity and Lagrange parameter the Lagrangian is changed in a way that the bound for the velocity (in relativity the speed of light) is incorporated in the model.

In the electromagnetic model, the Lagrangian would have to restrict the intensity of the fields. This was the idea which Borne and Infeld published already at the beginning of the last century. For the resulting system it is straightforward to calculate the fields for point charges. But unfortunately it is impossible to add the fields for several point charges (no superposition principle) since the resulting theory (and the PDE) are nonlinear. Physically this expresses, that the point charges do not act independently from each other but it accounts for certain interaction between the charges. Probably this interaction is really only important if charges are near enough to each other and locally it should be only influenced by the charge nearest. But it has not been possible to prove that up to now.

The electrostatic case is elliptic but has a singularity at each point charge. So no classical regularity results are directly applicable. On the other hand, there is an interesting interplay with geometry since the PDE occurs as the mean curvature equation of hypersurfaces in the Minkowski space in relativity.

The evolution problem is completely open. In the static case we have existence and uniqueness without really looking at the PDEs from the way the system is built. The PDE should provide at least qualitative information on the electric field. So if, e.g., there is a positive charge there could be a maximum of the field (for negative charges a minimum - respectively), and we would expect the field to be smooth outside these singular points. So a Lipschitz regular solution would seem probable. But it is open how to prove this mathematically.

A special property is that the model has infinitely many inherent scales, namely all even powers of the gradient of the field. So to understand maybe asymptotic limits in theses scales could be a first interesting step.

Denis Bonheure got his mathematical education at the Free University of Brussels and is working there as Professor of Mathematics at the moment.

• M. Kiessling: Electromagnetic Field Theory Without Divergence Problems 1, The Born Legacy, Journal of Statistical Physics, Volume 116, Issue 1, pp 1057-1122, 2004.
• M. Kiessling: Electromagnetic Field Theory Without Divergence Problems 2, A Least Invasively Quantized Theory, Journal of Statistical Physics, Volume 116, Issue 1, pp 1123-1159, 2004.
• M. Kiessling: On the motion of point defects in relativistic fields, in Quantum Field Theory and Gravity, Conceptual and Mathematical Advances in the Search for a Unified Framework, Finster e.a. (ed.), Springer, 2012.
• Y. Brenier: Some Geometric PDEs Related to Hydrodynamics and Electrodynamics, ICM Vol. III pp 761--772, 2002.

Nanophotonics is one great path into our future since it renders possible to build e.g. absorber, emitter or amplifier on a scale of a few dozen nanometers. To use this effectively we will have to understand firstly the resonances of plasmons and secondly the interaction of electromagnetic waves with complex media. Here on the one hand we can model light as waves and describe what is happening for the different frequencies of monochromatic light waves. We have to model the evolution in air or in more complex media.

On the other hand - taking the more particle centered point of view - we can try to model the reaction of the photons to certain stimuli.

The modelling is still in progress and explored in many different ways.

The main focus of our guest Claire Scheid who is working on nanophotonics is to solve the corresponding partial differential equations numerically. It is challenging that the nanoscale-photons have to be visible in a discretization for a makro domain. So one needs special ideas to have a geometrical description for changing properties of the material. Even on the fastest available computers it is still the bottleneck to make these computations fast and precise enough.

A special property which has to be reflected in the model is the delay in response of a photon to incoming light waves - also depending on the frequency of the light (which is connected to its velocity- also known as dispersion). So an equation for the the evolution of the electron polarization must be added to the standard model (which is the Maxwell system).

One can say that the model for the permeability has to take into account the whole history of the process. Mathematically this is done through a convolution operator in the equation. There is also the possibility to explain the same phenomenon in the frequency space as well.

In general the work in this field is possible only in good cooperation and interdisciplinary interaction with physicists - which also makes it especially interesting.

Since 2009 Claire Scheid works at INRIA méditerranée in Sophia-Antipolis as part of the Nachos-Team and is teaching at the university of Nice as a member of the Laboratoire Dieudonné.

She did her studies at the Ecole Normale Superieure in Lyon and later in Paris VI (Université Pierre et Marie Curie). For her PhD she changed to Grenoble and spent two years as Postdoc at the university in Oslo (Norway).

• R. Léger, J. Viquerat, C. Durochat, C. Scheid and S. Lanteri: A parallel non-confoming multi-element DGTD method for the simulation of electromagnetic wave interaction with metallic nanoparticles, J. Comput. Appl. Math. Vol 270, p. 330-342, 2014.
• S. Descombes, C. Durochat, S. Lanteri, L. Moya, C. Scheid, J. Viquerat: Recent advances on a DGTD method for time-domain electromagnetism, Photonics and Nanostructures, Volume 11, issue 4, 291-302, 2013.
• K. Busch, M. König, J. Niegemann: Discontinuous Galerkin methods in nanophotonics, Laser and Photonics Reviews, 5, pp. 1–37, 2011.

To separate one single instrument from the acoustic sound of a whole orchestra- just by knowing its exact position- gives a good idea of the concept of wave splitting, the research topic of Marie Kray. Interestingly, an approach for solving this problem was found by the investigation of side-effects of absorbing boundary conditions (ABC) for time-dependent wave problems- the perfectly matched layers are an important example for ABCs.

Marie Kray works in the Numerical Analysis group of Prof. Grote in Mathematical Department of the University of Basel. She did her PhD 2012 in the Laboratoire Jacques-Louis Lions in Paris and got her professional education in Strasbourg and Orsay.

Since boundaries occur at the surface of volumes, the boundary manifold has one spatial dimension less than the actual regarded physical domain. Therefore, the treatment of normal derivatives as in the Neumann boundary condition needs special care.

The implicit Crank-Nicolson method turned out to be a good numerical scheme for integrating the time derivative, and an upwinding scheme solved the discretized hyperbolic problem for the space dimension.

An alternative approach to separate the signals from several point sources or scatterers is to apply global integral boundary conditions and to assume a time-harmonic representation.

The presented methods have important applications in medical imaging: A wide range of methods work well for single scatterers, but Tumors often tend to spread to several places. This serverely impedes inverse problem reconstruction methods such as the TRAC method, but the separation of waves enhances the use of these methods on problems with several scatterers.

• F. Assous, M. Kray, F. Nataf, E. Turkel: Time-reversed absorbing condition: application to inverse problems, Inverse Problems, 27(6), 065003, 2011.
• F. Assous, M. Kray, F. Nataf: Time reversal techniques for multitarget identification, in Ultrasonics Symposium (IUS), IEEE International (pp. 143-145). IEEE, 2013.
• M. Grote, M. Kray, F. Nataf, F. Assous: Wave splitting for time-dependent scattered field separation, Comptes Rendus Mathematique, 353(6), 523-527, 2015.

Prof. Francisco Sayas from the Department of Mathematical Sciences of the University of Delaware in Newark has been visiting our faculty in June 2015. He is an expert in the simulation of scattering of transient acoustic waves.

Scattering is a phenomenon in the propagation of waves. An interesting example from our everyday experience is when sound waves hit obstacles the wave field gets distorted. So, in a way, we can "hear" the obstacle.

Sound waves are scalar, namely, changes in pressure. Other wave types scatter as well but can have a more complex structure. For example, seismic waves are elastic waves and travel at two different speeds as primary and secondary waves.

As mathematician, one can abandon the application completely and say a wave is defined as a solution of a Wave Equation. Hereby, we also mean finding these solution in different appropriate function spaces (which represent certain properties of the class of solutions), but it is a very global look onto different wave properties and gives a general idea about waves. The equations are treated as an entity of their own right. Only later in the process it makes sense to compare the results with experiments and to decide if the equations fit or are too simplified.

Prof. Sayas startet out in a "save elliptic world" with well-established and classical theories such as the mapping property between data and solutions. But for the study of wave equations, today there is no classical or standard method, but very many different tools are used to find different types of results, such as the preservation of energy. Sometimes it is obvious, that the results cannot be optimal (or sharp) if e.g. properties like convexity of obstacles do not play any role in getting results. And many questions are still wide open. Also, the numerical methods must be well designed.

Up to now, transient waves are the most challenging and interesting problem for Prof. Sayas. They include all frequencies and propagate in time. So it is difficult to find the correct speed of propagation and also dispersion enters the configuration. On the one hand, the existence and regularity together with other properties of solutions have to be shown, but on the other hand, it is necessary to calculate the propagation process for simulations - i.e. the solutions - numerically.

There are many different numerical schemes for bounded domains. Prof. Sayas prefers FEM and combines them with boundary integral equations as representative for the outer domain effects. The big advantage of the boundary integral representation is that it is physical correct but unfortunately, it is very complicated and all points on the boundary are interconnected.

Finite Elements fit well to a black box approach which leads to its popularity among engineers. The regularity of the boundary can be really low if one chooses Galerkin methods. The combination of both tools is a bit tricky since the solver for the Wave Equations needs data on the boundary which it has to get from the Boundary element code and vice versa. Through this coupling it is already clear that in the coding the integration of the different tools is an important part and has to be done in a way that all users of the code which will improve it in the future can understand what is happening.

Prof. Sayas is fascinated by his research field. This is also due to its educational aspect: the challenging mathematics, the set of tools still mainly unclear together with the intensive computational part of his work. The area is still wide open and one has to explain mathematics to other people interested in the results.

In his carreer he started out with studying Finite Elements at the University in Zaragoza and worked on boundary elements with his PhD-supervisor from France. After some time he was looking for a challenging new topic and found his field in which he can combine both fields. He has worked three years at the University of Minnesota (2007-2010) and decided to find his future at a University in the U.S.. In this way he arrived at the University of Delaware and is very satisfied with the opportunities in his field of research and the chances for young researchers.

• deltaBEM - Easy to Implement Boundary Integral Equations, open source software developed by Team Pancho at the Department of Mathematical Sciences in Delaware.
• A. R. Laliena, F. J. Sayas: Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves, Numerische Mathematik, 112(4), 637-678, 2009.
• F. J. Sayas: Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations, Numerische Mathematik, 124(1), 121-149, 2013.
• Modellansatz Podcast 003: Unsichtbarkeit (in German)

Prof. Enrique Zuazua is a Distinguished Professor of Ikerbasque (Basque Foundation for Science) and Founding Scientific Director at the Basque Center for Applied Mathematics (BCAM), which he pushed into life in 2008.

He is also Professor in leave of Applied Mathematics at the Universidad Autónoma de Madrid (UAM) and a Humboldt Awardee at the University of Erlangen-Nuremberg (FAU) as well. He was invited by the PDE-group of our Faculty in Karlsruhe to join our work on Wave Phenomena for some days in May 2015.

In our conversation he admits that waves have been holding his interest since his work as a PhD student in Paris at the Université Pierre-et-Marie-Curie in the world famous group of Jacques-Louis Lions.

Indeed, waves are everywhere. They are visible in everything which vibrates and are an integral part of life itself. In our work as mathematician very often the task is to influence waves and vibrating structures like houses or antennae such that they remain stable. This leads to control problems like feedback control for elastic materials.

In these problems it is unavoidable to always have a look at the whole process. It starts with modelling the problem into equations, analysing these equations (existence, uniqueness and regularity of solutions and well-posedness of the problem), finding the right numerical schemes and validating the results against the process which has been modelled. Very often there is a large gap between the control in the discrete process and the numerical approximation of the model equations and some of these differences are explainable in the framework of the theory for hyperbolic partial differential equations and not down to numerical or calculation errors.

In the study of Prof. Zuazua the interaction between the numerical grid and the propagation of waves of different frequencies leads to very intuitive results which also provide clear guidelines what to do about the so-called spurious wave phenomena produced by high frequencies, an example of which is shown in this podcast episode image.

This is an inherent property of that sort of equations which are able to model the many variants of waves which exist. They are rich but also difficult to handle. This difficulty is visible in the number of results on existence, uniqueness and regularity which is tiny compared to elliptic and parabolic equations but also in the difficulty to find the right numerical schemes for them. On the other hand they have the big advantage that they are best suited for finding effective methods in massively parallel computers. Also there is a strong connection to so-called Inverse Problems on the theoretical side and through applications where the measurement of waves is used to find oil and water in the ground, e.g (see, e.g. our Podcast Modell004 on Oil Exploration).

Prof. Zuazua has a lot of experience in working together with engineers. His first joint project was shape optimization for airfoils. The geometric form and the waves around it interact in a lot of ways and on different levels. Also water management has a lot of interesting and open questions on which he is working with colleagues in Zaragoza. At the moment there is a strong collaboration with the group of Prof. Leugering in Erlangen which is invested in a Transregio research initiative on gasnets which is a fascinating topic ranging from our everyday expectations to have a reliable water and gas supply at home to the latest mathematical research on control. Of course, in working with engineers there is always a certain delay (in both directions) since the culture and the results and questions have to be translated and formulated in a relevant form between engineers and mathematicians.

In dealing with theses questions there are two main risks: Firstly, one finds wrong results which are obviously wrong and secondly wrong results which look right but are wrong nonetheless. Here it is the crucial role of mathematicians to have the right framework to find these errors.

Prof. Zuazua is a proud Basque. Of the 2.5 Mill. members of the basque people most are living in Spain with a minority status of their culture and language. But since the end of the Franco era this has been translated into special efforts to push culture and education in the region. In less than 40 years this transformed the society immensely and led to modern universities, relevant science and culture which grew out of "nothing". Now Spain and the Basque country have strong bonds to the part of Europe on the other side of the Pyrenees and especially with industry and research in Germany. The Basque university has several campuses and teaches 40.000 students. This success could be a good example how to extend our education system and provide possibilities for young people which is so much a part of our culture in Europe across the boundaries of our continent.