(September 22) Jing-Mei Qiu: Conservative high order semi-Lagrangian WENO method for the Vlasov equation — We propose a novel semi-Lagrangian method for the Vlasov equation, which combines Strang splitting in time with WENO reconstructions in space. A key insight is that the spatial interpolation matrices, used in the reconstruction process can be factored into flux matrices, because of which WENO can be applied. The CFL time step restriction is removed in the semi-Lagrangian framework. The quality of the method is demonstrated by several classical problems in plasma physics.

(September 29) Elisabeth Ullmann: Preconditioning Stochastic Galerkin Saddle Point Systems — We study efficient iterative solvers for Galerkin equations associated with mixed finite element discretizations of second-order elliptic partial differential equations (PDEs) with random coefficient functions. Such systems arise, for example, from discretized Darcy flow problems with random permeability coefficients. The Galerkin matrix has a familiar saddle point structure, however, due to the coupling of (standard) mixed finite element discretizations in the physical space and global polynomial chaos approximations on a probability space, the number of unknowns is huge. Moreover, the leading blocks of the saddle point matrix are sums of Kronecker products of pairs of matrices associated with the physical and stochastic discretization, respectively. Depending on the models employed for the random coefficient function, this block can be block-dense, and the cost of a matrix-vector product is non-trivial.

We analyze block-diagonal preconditioners of Schur complement and augmented type for use with MINRES. First, we consider an inexpensive mean-based preconditioner based on fast solvers for scalar diffusion problems. For stochastically linear random coefficient functions, which arise, for example, from a truncated Karhunen-Love expansion, with moderate fluctuations in the data relative to the mean value, we demonstrate that this gives an efficient way for solving the large coupled Galerkin system.

If, on the other hand, the random diffusion coefficient is a lognormal random field approximated via a nonlinear function of random parameters, mean-based preconditioners are not effective. For this case, we combine so-called Kronecker product preconditioners with the Schur complement and augmented preconditioning approach, respectively. We study the spectral properties of the preconditioned saddle point matrix and demonstrate numerically the improved robustness of the Kronecker product preconditioners compared to the mean-based approach with respect to key statistical parameters of the random diffusion coefficient.

This is joint work with Catherine Powell, David Silvester (Manchester, UK) and Oliver Ernst (Freiberg, Germany).

(October 13) Tom Hughes: Variational Multiscale Methods in Computational Fluid Dynamics — I will discuss the development of the variational multiscale (VMS) approach for solving partial differential equation systems arising in fluid dynamics. The background is this: Any reasonable method utilizing functions capable of resolving the exact solution will obtain it. However, in numerical analysis we typically employ finite-dimensional spaces of functions that are unable to give good approximations for many fluid dynamical problems of practical interest. In addition, the basic variational methods (e.g., Galerkin) are not as reasonable as often assumed in the finite-dimensional setting. Stability, present in the continuous setting, is often not inherited for typically utilized finite-dimensional subspaces. The possibilities are to improve the function spaces, improve the variational methods, or both. Improving the spaces is possible but difficult. It has been a pathway to attaining stability for simpler problems. For more complex problems, enhancing the stability of the variational method, without upsetting its consistency, has been a more practical direction. This is the essence of so-called stabilized methods. But stability is not the only issue in computational fluid dynamics (CFD). In modeling turbulence, the effects of unresolved scales on resolved scales must also be accounted for.

VMS is a paradigm that derives directly from the variational formulation of the partial differential equations. In very simple situations it coincides with stabilized methods, but in more complicated cases it is richer. In addition to providing additional stability, it accounts for the effects of unresolved scales, and thus is a general framework for turbulence modeling as well as the derivation of CFD methods.

Recent research in VMS concerns comparisons with stabilized methods, calculation of the fine-scale Greens function, the use of continuous and discontinuous function spaces, minimizing the error in various measures, approximating discontinuities, the fine-scale field as error estimator, geometrically inspired approximations, weak boundary conditions, and turbulence modeling. In my talk I will sample from some of these recent developments and emphasize the use of VMS as a theoretical means for developing turbulence models. In particular, I will present a formulation of LES that is derived entirely from the Navier-Stokes equations without recourse to any external ad hoc devices, such as eddy viscosity models, and I will demonstrate the effectiveness of the ideas through numerical examples.

(October 20) Pino Martin: Numerical Challenges for Direct and Large-Eddy Simulations of Highly Compressible Turbulence — The detailed simulation of compressible turbulent flows requires solving the conservation of mass, momentum and energy equations. For direct numerical simulations (DNS) and large eddy simulations (LES), a wide range of possible turbulent length scales and time scales must be resolved by the numerical method. Thus, DNS and LES require accurate representation of time-dependent propagation of high wave number (or high frequency), small amplitude waves. In addition, compressible turbulent flows are characterized by shockwaves that result in a sudden change of the fluid properties. Strong fluctuations can lead to the formation of transient shocklets, and boundary conditions and flow geometry might result in stronger, more permanent shocks. Therefore, methods for compressible turbulent flows require robust shock capturing, as well as minimal numerical dissipation and dispersion errors. We have worked on refashioning weighted essentially non-oscillatory methods for the simulation of compressible turbulence. This work has enabled efficient and accurate DNS of turbulent flows that are relevant to reusable launch vehicles and scramjet engines.

In our current work, we are using converged DNS data of canonical flows, such as highly compressible isotropic turbulence interacting with shock waves, and DNS data of shock wave and boundary layer interactions to assess and develop a robust and efficient LES methodology for this type of flows. The formulation of LES requires the use of filtering operations. We are exploring a class of shock-confining filters (SCF) to avoid filtering across shocks. We find that linear filtering consistently causes data to exhibit anomalies immediately downstream of the main shock and that these anomalies are avoided by the application of SCF.

In this talk, I will demonstrate the reduction of numerical dissipation given by using linearly and non-linearly optimized WENO methods, and how the adaptation mechanism of WENO prevents a further reduction of the numerical dissipation. In addition, I will illustrate the need for nonlinear filtering procedures to enable robust LES of compressible turbulence.

(October 27) Emmanuil H. Georgoulis: Discontinuous Galerkin Methods for Convection-Reaction-Diffusion systems with Transmission Conditions — We propose a family of Interior Penalty Discontinuous Galerkin (IP-DG) finite element methods for the solution of convection-reaction-diffusion systems on partitioned subdomains. Kadem-Katchalsky-type [Biochim Biophys Acta, Volume 27, 1958] transmission conditions are imposed at the sub-domain interfaces. The problem considered is relevant to the modeling of chemical species passing through thin biological membranes. In particular, we consider applications in cellular signal transduction.

The reasoning of proposing discontinuous Galerkin (DG) methods for this class of problems is threefold. First, DG methods have good stability properties in regions where the convection is the dominant feature of the problem (without the need of additional streamline diffusion-type stabilisation). Second, DG methods for parabolic problems have some attractive features such as good local conservation of the state variables and produce block-diagonal mass matrices. Finally, they are particularly suited to the discretization of transmission conditions; indeed, the transmission conditions can be incorporated within an existing DG code by the addition of appropriate elemental boundary terms in the bilinear form.

We show that the optimal error analysis of the IP-DG method by Suli and Lasis [SIAM J Numer Anal, 45 no. 4 (2007)] can be extended to semi-linear convection-reaction-diffusion systems with nonlinear transmission conditions under appropriate assumptions on the reaction terms growth. A series of numerical experiments highlight the good stability and accuracy properties of the proposed method. This is joint work with Andrea Cangiani (Milano-Bicocca, Italy) and Max Jensen (Durham, UK).

(October 28) Yekaterina Epshteyn: Chemotaxis and Numerical Methods for Chemotaxis Models — In this work, first we will discuss several chemotaxis models including the classical Keller-Segel model. Chemotaxis is the phenomenon in which cells, for example bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment. The mathematical models of chemotaxis are usually described by highly nonlinear time dependent systems of PDEs. Therefore, accurate and efficient numerical methods are very important for the validation and analysis of these systems. Furthermore, it is known that the solutions of chemotaxis models may blow up or may exhibit very singular spiky behavior. Capturing such solutions numerically is a challenging problem. In our work we propose a family of new high-order interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model with parabolic-parabolic coupling. As it can be shown the convective part of this model is of a mixed hyperbolic-elliptic type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of the proposed methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original Keller-Segel model in the form of a convection-diffusion-reaction system. We then design interior penalty discontinuous Galerkin methods (IP) for the rewritten Keller-Segel system. Our methods employ the central-upwind numerical fluxes, originally developed in the context of finite-volume methods for hyperbolic systems of conservation laws. We prove error estimates for the proposed high-order discontinuous Galerkin schemes. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution. Some numerical experiments to demonstrate the stability and high accuracy of the proposed methods and comparison with other methods will be presented. Ongoing research projects will be discussed as well.

(November 3) Andrea Bonito: Numerical Approximation of the Time-Harmonic Maxwell System: Rehabilitation of the Lagrange Finite Elements — We describe a new approximation technique for the Maxwell eigenvalue problem using $\mathbf{H}^1$-conforming finite elements.

We start our discussion by recalling relevant properties of the Maxwell operator emphasizing the difficulties for $\mathbf{H}^1$-conforming finite elements to deliver a correct spectral approximation. In fact, they were commonly believed not to be able to provide such approximation.

The key idea consists of controlling the divergence of the electric field in a fractional Sobolev space $H^{-\alpha}$ with $\alpha\in (\frac12,1)$. We examine first a non-implementable scheme to explain the essense of the method. We next consider the extension to the actual method used in practice, and finally conclude with numerical experiments.

This is joint work with J.-L. Guermond.

(November 10) Julianne Chung: Numerical Methods for Large-Scale Ill-Posed Inverse Problems — Many scientific and engineering applications require numerical methods to compute efficient and reliable solutions to inverse problems. The basic goal of an inverse problem is to compute an approximation of the original model, given observed data and knowledge about the forward model. Physical systems that require reconstruction of an unknown input signal from the measured output signal are natural examples of inverse problems. In other systems, the internal structure of an object is desired, but only measured output data is provided. The difficulty with ill-posed inverse problems is that small errors may give rise to significant errors in the computed approximations, so regularization must be used to compute stable solution approximations. Furthermore, real-life applications often require the computer to process extremely large amounts of data, and previously proposed methods for solving inverse problems are not adequate for these large-scale problems.

In this talk, we investigate hybrid methods for regularization of linear and nonlinear least squares problems and describe an efficient parallel implementation based on the Message Passing Interface (MPI) library for use on state-of-the-art computer architectures. Regularization for nonlinear Poisson based models, such as those arising from digital tomosynthesis reconstruction, is significantly more challenging. However, reconstruction algorithms for polyenergetic tomosynthesis will be discussed, and numerical experiments illustrate the effectiveness and efficiency of the proposed methods.

(November 16) Weiqin Ren: A seamless multiscale method and its application to complex fluids — I will present a seamless multiscale method for the study of multiscale problems. The multiscale method aims at capturing the macroscale behavior of a given system which is modeled by a (incomplete) macroscale model. In the multiscale method, the macro model is coupled with a micro model: The macro model provides the necessary constraint for the micro model and the micro model supplies the missing data (e.g. the constitutive relation or the boundary conditions) needed in the macro model. In the multiscale method, the two models evolve simultaneously using different time steps, and they exchange data at every step. The micro model uses its own appropriate (micro) time step. The macro model uses a macro time step but runs at a slower pace than required by accuracy and stability considerations in order for the micro dynamics to have sufficient time to adapt to the environment provided by the macro state. The method has the advantage that it does not require the reinitialization of the micro model at each macro time step. The data computed from the micro model is implicitly averaged over time. In this talk, I will discuss the algorithm of the multiscale method, the error analysis, and its application to complex fluids. I will also briefly discuss the stability analysis of different coupling schemes in domain-decomposition type of multiscale methods.

(November 23) Maria Cameron: The MaxFlux Functional: Derivation, Numerics, and Application to LJ-38 — The overdamped Langevin equation is often used as a model in molecular dynamics. At low temperatures, a system evolving according to such an SDE spends most of the time near the potential minima and performs rare transitions between them. A number of methods have been developed to study the most likely transition paths. I will focus on one of them: the MaxFlux functional.

The MaxFlux functional has been around for almost thirty years but not widely used because it is challenging to minimize. Its minimizer provides a path along which the reactive flux is maximal at a given finite temperature. I will show two ways to derive it in the framework of transition path theory: the lower bound approach and the geometrical approach. I will present an efficient way to minimize the MaxFlux functional numerically. I will demonstrate its application to the problem of finding the most likely transition paths in the Lennard-Jones-38 cluster between the face-centered-cubic and icosahedral structures.

(November 24) Yanzhi Zhang: Analysis and Simulation of Bose-Einstein Condensation & Quadrature-rule Type Approximations to the Quasicontinuum Method — This talk includes two parts:

Part I: Since its first observation in 1995, Bose-Einstein condensation (BEC) has been a popular topic. Recently, quantized vortices in BEC have attracted lots of attention from researchers. The stationary states are studied in the Thomas-Fermi regime using an asymptotic method. The dynamical laws are derived for the dynamics of quantized vortices, and other properties of the condensates are also discussed. Efficient and accurate numerical methods are developed and applied to verify the analytical properties. The structures of vortex lattices in both 2D and 3D are numerically studied, and the generation of these vortex lattices is also presented. In addition, the interactions between a vortex pair and a vortex dipole are investigated.

Part II: Quasicontinuum methods using representative particles provide a simplified model to study huge molecular systems. The objective of this study is to develop quadrature-rule type approximations to further simplify the quasicontinuum method. For both short and long-range interatomic interactions, the complexity of this method depends on the number of representative particles but not on the total number of particles. Numerical experiments and complexity estimates illustrate that the quadrature-rule type method preserves much of accuracy of the quasicontinuum method, but at a much lower cost.

(December 1) Anil Zenginoglu: Solving the outer boundary problem using hyperboloidal compactification — In numerical calculations of hyperbolic partial differential equations, one typically truncates the solution domain by introducing an artificial outer boundary. This boundary is, in general, not part of the original problem, which introduces difficulties related to the construction of transparent (or absorbing) boundary conditions. I will present a clean and geometric solution to the outer boundary problem based on compactification along hyperboloidal surfaces. These surfaces are well-adapted to study outgoing characteristicts. To demonstrate how well the method works in practice, I will discuss the example of the cubic wave equation in flat spacetime. In joint work with Piotr Bizon, we recently discovered a universal attractor in the solution space of the cubic wave equation that encompasses both global and blow-up solutions including an explicit description of the critical behavior near the threshold of blow-up. I will show how hyperboloidal compactification was instrumental in unravelling the dynamics of the cubic wave equation. Finally, I will point out some applications of hyperboloidal compactification in general relativity.

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