Lattice Boltzmann methods (LBM) provide a novel technique for simulating systems that are governed by dissipative and dispersive nonlinear partial differential equations. The method avoids the direct treatment of computationally expensive nonlinear convective derivatives by advectively streaming mesoscopic particle distribution functions, which describe the macroscopic system. LBMs that use the linear BGK collision operator produce simple algorithms, which lend themselves to ideal parallelization on platforms containing multiple processing environments. Moreover, LBMs can incorporate most of the modern computational techniques used in finite difference simulations, including non-uniform and adaptive grids. I will present a simple development of LBMs from kinetic theory and will show how the particle distribution functions can be chosen to describe resistive magnetohydrodynamics (MHD). I will then present some simulations of one and two dimensional resistive MHD.

An effort of combining theoretical/computational analyses and protein engineering methods has been made to probe the folding mechanism of a modular protein, SH3, utilizing an Energy Landscape Theory and novel mutagenetic phi-value analysis. Particularly emphasis was given to core residues and the effect of desolvation during the protein folding event. Experimentally that was probed by replacing the core valines by isosteric threonines. These mutations have the advantage of keeping core structurally invariant while affecting the core stability relative to the unfolded state. Although the valines that form the core appear spatially invariant, the folding kinetics of their threonine mutants varies, indicating their different extent of solvation in the transition state ensemble. Theoretical and computational studies predicted the distribution of folding kinetics of threonine mutants without previous knowledge of the measured rates. This initial success encourages further investigations of providing molecular details behind these macroscopic phenomena and on the role of solvation in the folding reaction.

Note: Special Time and place

In this talk, We investigate the vortex nucleation in a thin superconducting hollow sphere. The problem is studied using a simplified system of Ginzburg-Landau equations. We present finite volume methods which preserve the discrete gauge invariance for the time dependent simulation. The spatial discretization is based on a spherical centroidal Voronoi tessellation which offers a very effective high resolution mesh on the sphere for the order parameter as well as other physically interesting variables such as the super-current and the induced magnetic field. Various vortex configurations and energy diagrams are computed.

Nonconvex minimisation problems are encountered in many applications such as phase transitions in solids (1) or liquids but also in optimal design tasks (2) or micromagnetism (3). In contrast to rubber-type elastic materials and many other variational problems in continuum mechanics, the minimal energy may be not attained. In the sense of (Sobolev) functions, the non-rank-one convex minimisation problem ($M$) is ill-posed: As illustrated in the introduction of FERM, the gradients of infimising sequences are enforced to develop finer and finer oscillations called microstructures. Some macroscopic or effective quantities, however, are well-posed and the target of an efficient numerical treatment. The presentation proposes adaptive mesh-refining algorithms for the finite element method for the effective equations ($R$), i.e. the macroscopic problem obtained from relaxation theory. For some class of convexified model problems, a~priori and a~posteriori error control is available with an reliability-efficiency gap. Nevertheless, convergence of some adaptive finite element schemes is guaranteed. Applications involve model situations for (1), (2), and (3) where the relaxation is provided by a simple convexification.

One of the useful models in the study of turbulent convection is the so-called infinite Prandtl number model which is derived by formally setting the Prandtl number equal to infinity in the Boussinesq approximation of Rayleigh-Benard convection. The model is particularly relevant for fluids with large Prandtl number such as the earth's mantle, silicone oil, and many gases under high pressure. In this talk I will present a few results in the systematic study of the behavior of solutions to the Boussinesq system at large Prandtl number. We first establish the validity of the infinite Prandtl number model as an approximation of the Boussinesq system at large Prandtl number on any finite but fixed time interval. Such an approximation is singular involving an initial transition layer. We then argue that individual trajectories of the Boussinesq system are not expected to remain close to those of the infinite Prandtl number model over a long period of time due to instability. The validity of the infinite Prandtl number model over long time interval is then studied in terms of the proximity of invariant measures (stationary statistical solutions) and global attractors which are commonly used in the study of long time behaviors. Finally we study the long time behavior of the infinite Prandtl number model. Some numerical issues will also be discussed.

Prediction of a pollution transport in flows is an important problem in many industrial
and environmental projects. Different mathematical models are used to describe the propagation of the pollutant and to obtain its accurate location and concentration.

We will consider the flow modeled by the Saint-Venant system of shallow water equations while the pollutant propagation is described by a transport equation. Designing an accurate, efficient and reliable numerical method for this model is a challenging task: solutions are typically nonsmooth, they may contain both nonlinear shock and rarefaction waves, and linear discontinuities in the pollution concentration. Moreover, the interaction with a nonflat bottom may result in very complicated wave structures and nontrivial equilibrium, which are hard to preserve numerically. In addition, dry states (arising, for example, in dam break problems) need special attention, since (even small) numerical oscillations may lead to nonphysical negative values of the water depth there.

I will  present a new hybrid numerical method for computing the transport of a passive pollutant by a flow. The idea behind the new finite-volume-particle method is to use different schemes for the flow and pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilized at the right place. This results in a significantly enhanced resolution of the computed solution.

This is joint work with A. Kurganov and G. Petrova.

Although the wealth and complexity of phenomena in cell biology have raised considerable interest among theorists the impact of theoretical work on scientific progress in this discipline has been very limited. I want to discuss two main challenges theorists face when entering the world of biological phenomena: First, many of the scientific procedures and tools which have proven very useful in the physical sciences and other classical disciplines of applied mathematics fail or do not yet exist in computational biology. For example do most biological systems not yield themselves readily to rigorous reductionism. On the other hand are approaches to solve the problem of biological complexity through the development of effective theories (akin to the transition from mechanics to statistical mechanics and finally thermodynamics) still missing. The second challenge results from the difficulty of information exchange between biologists and theorists. Scientists with theoretical background frequently lack the biological knowledge needed to understand the work of experimentalists while biologists are not trained in presenting their research in abstract terms.The modeling and simulation software we are currently developing is an attempt to address some of these problems. It allows its users (mainly biologists) to define molecular properties and mechanisms of cellular behavior in much biological detail through a graphical interface. It then transforms the biological models into a format which makes it possible to analyze the structural properties of the model or to simulate the model system by numerically integrating reaction-diffusion equations.

In this talk we review mathematical results on the 3D Navier-Stokes and Euler Equations with initial data characterized by uniformly large vorticity. We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for a class of large initial data both in R3 and bounded cylindrical domains; as well as long time existence of smooth solutions for 3D incompressible Euler Equations. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Navier-Stokes equations. We establish global regularity of the latter without any restriction on the size of 3D initial data and with the help of strong convergence theorems bootstrap this into the global regularity of the weak solutions of 3D Navier-Stokes Equations with weakly aligned uniformly large vorticity at t=0. For the 3D incompressible Euler Equations with initial data characterized by uniformly large vorticity which induce large vortex stretching in bounded cylindrical domains we prove existence on arbitrary large time intervals of regular solutions with large kinetic energy; the ratio of the large enstrophy to the large kinetic energy is of order one.

Note Special Day: Monday November 24

A method for numerical investigation of nonlinear wave dynamics based on direct hydrodynamical modeling of 1-D potential periodic surface waves is developed. By a nonstationary conformal mapping, the principal equations are rewritten in a surface-following coordinate system and reduced to two simple evolutionary equations for the elevation and the velocity potential of the surface; Fourier expansion is used to approximate these equations. High accuracy of the method is confirmed (i) by control of main integral invariants, (ii) by validation of the nonstationary model against stationary solutions (Stokes? Crappers? and gravity-capillary waves in a moving coordinate system) and (ii) by comparison between the results obtained with different resolution in the horizontal. A number of long-term simulations of gravity, gravity-capillary and pure capillary waves with various initial conditions, were performed; for the simulated wave fields, distribution of energy and phase speed over full spectra were analysed. Numerical experiments with initially monochromatic waves with different steepness show that the model is able to simulate breaking conditions when the surface becomes a multi-valued function of the horizontal coordinate; an estimate of the critical initial wave height that divides between non-breaking and eventually breaking waves is obtained. Simulations of nonlinear evolution of a wave field represented initially by two modes with close wave numbers (amplitude modulation) and a wave field with a phase modulation both result in appearance of large and very steep waves, which also break if the initial amplitudes are sufficiently large. The method developed may be applied to a broad range of problems where the assumption of one dimensionality is acceptable. The problem of numerical simulation of 2-D potential and 3-D non-potential waves options is also briefly discussed.

We modelize the motion of particles in a fluid as the coupling of a fluid equation and a Fokker-Planck type kinetic equation. This system takes into account both the Brownian effect and the interactions between the fluid and the particles. Such system is very costly in a numerical point of view. We investigate how we can derive a simpler model and show mathematically the convergence from the previous model to the simpler one.