The literature for convergence analysis of numerical approximations of hyperbolic equations is vast and contain many highly elegant results. Still most of today's simulations of nonlinear problems lack a rigorous foundation. We will discuss the history and focus on some inherent difficulties with a view to the future.

Moment Equations are used as non-linear approximations for the solution of Boltzmann's equation in kinetic gas theory. As a continuum model of gas flows they can successfully replace classical gas dynamical equations when the thermal non-equilibrium is too strong. The approximation theory must be build on accuracy with respect to the Boltzmann equation but also need to satisfy structural requirements like the existence of an entropy law. We will discuss recent developments how to establish moment equations with entropy law.

The Euler-Maxwell equations are a moment approximation of the kinetic Vlasov-Maxwell description. These equations arise in the plasma physics modeling of plasma rockets, various fusion devices, lasers, and astrophysics. Multi-fluid Euler-Maxwell approximations have received considerable attention because of the excellent prediction of magnetic reconnection in low viscosity plasma regimes [Loverich and Shumlak (2005), Johnson and Rossmanith, (2008)]. Note that standard magnetohydrodynamics (MHD) approximations can be obtained as simplifications of the Euler-Maxwell plasma system. Even so, the differences between Euler-Maxwell and MHD have a profound effect on the theory and construction of numerical methods for each system.

In this presentation, we use the multi-fluid Euler-Maxwell system to illustrate and review the construction of energy stable discontinuous Galerkin approximations for nonlinear hyperbolic systems. Surprisingly, the stability theory underpinning these numerical approximations is significantly simpler for Euler-Maxwell systems when conpared to MHD!! This occurs due to the different way solenoidal constraints (involutions) enter into the energy balance for each system. In addition, energy stable boundary conditions (e.g. conducting walls) emerge naturally from the formulation. Finally, we show calculations of 2-fluid Euler-Maxwell plasmas undergoing magnetic reconnection to assess properties of the numerical methods that we have implemented.

Several mathematical models are described by hyperbolic systems with stiff relaxation, defined by a small relaxation time ϵ. In the so-called hyperbolic relaxation, as the ϵ vanishes, the system relaxes to another hyperbolic system with smaller number of equations. Classical method of line approach based on finite difference schemes present the double difficulty: the stiffness of the source term, and the requirement that in the limit of infinite stiffness the scheme becomes a consistent discretization of the relaxed system. Implicit-Explicit (IMEX) schemes constitute a powerful tool to attach this problem (see for example [4] and references therein). In such schemes, the (non stiff) hyperbolic terms are treated explicitly, while the (stiff) source terms are treated implicitly. It will be shown that under certain conditions, IMEX schemes provide the correct asymptotic behavior as the stiffness parameter ϵ vanishes. Furthermore, singular perturbation method can be employed to derive order conditions which will provide uniform accuracy in ϵ [1]. In the case of diffusive relaxation the situation is much more complicated. In this case the characteristic speeds of the hyperbolic part depend on ϵ and diverge as ϵ→0. There are methods that allow to overcome such stiffness, and that allow the construction of asymptotic preserving shemes that, in the limit of infinite stiffness, reduce to a consistent explicit scheme for the underlying diffusion (or convection-diffusion) equation [3]. Here we present two techniques for the construction of IMEX schemes which capture the diffusive limit, without the classical stability restriction on the time step ∆t = O(∆x²). The first one, developed in collaboration with L.Pareschi, is based on an implicit treatment of some hyperbolic terms, which can however be explicitly computed, while the second one, which treats the hyperbolic terms explicitly, is obtained by applying additional condition on the RK coefficients, obtained by a singular perturbation expansion of both the solution and the numerical solution [2]. Several numerical tests will be presented that illustrate the robustness and generality of the methods. Most of the results presented have been developed in collaboration with Sebastiano Boscarino, DMI, University of Catania.

[1] Boscarino S.: Error Analysis of IMEX Runge-Kutta Methods Derided from Differential Alge- braic Systems, SIAM J. Numer. Anal. 45, No 4, 1600-1621 (2007).

[2] Boscarino S., Russo G., Pareschi L.: IMEX Runge-Kutta schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, in preparation.

[3] Naldi G., Pareschi L.: Numerical Schens for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation. SIAM J. on Numer. Anal., 37, 1246-1270 (2000).

[4] Pareschi L., Russo G.: Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxations. Journal of Scientific Computing 25, 129-155 (2005).

We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws containing stiff source term. On one hand, we determine the relevant asymptotic expansion, derive a reduced system in the limit, and discuss the role of an entropy condition to establish the diffusive nature of the reduced system. On the other hand, we propose a new numerical scheme of finite volume type, which allows us to recover the correct asymptotic regime. The associated discrete form of the diffusion system is achieved via a suitable discretization compatible with the stiff source term. Our theoretical results are illustrated with several models from continuum physics and numerical experiments demonstrating the relevance of the proposed theory and numerical strategy. (This is a joint work with C. Berthon and R. Turpault.)

Non-conservative hyperbolic systems arise in a wide range of applications. The difficulty lies in the fact that shock relations depend not only on the immediate states ahead/behind the shock, but also on the viscous path that connects them. The mathematical theory of weak solutions has been generalized to the nonconservative setup us ng viscosity and viscous path. While advances have been made on the theoretical front, those advances have been slow to translate into successful numerical methods. In this talk, we shed light on some of the difficulties involved by considering the so-called path-conservative numerical scheme applied to an illuminating example from gas. This is joint work with Remi Abgrall, University of Bordeaux, France

Many geophysical flows over topography can be modeled by two-dimensional depth-averaged fluid dynamics equations. The shallow water equations are the simplest example of this type, though it is often necessary to incorporate non-hydrostatic pressures, more complicated rheologies (e.g. for avalanches, landslides, or debris flows), or to use multi-layer models, e.g. for capturing internal waves or to model a landslide-induced tsunamis.

These equations are generally hyperbolic and can be modeled using high-resolution finite volume methods designed for such problems. However, several features ew algorithmic challenges, such as the fact that the depth goes to zero at the edge of the flow and that vastly differing spatial scales must often be modeled, making adaptive mesh refinement essential. I will discuss some of these algorithms and the GeoClaw software, a specialized version of Clawpack that is aimed at solving real-world geophysical flow problems over topography. Inparticular I will show results from some recent tsunamis and potential future events. For information about the software, see www.clawpack.org/geoclaw.

We construct uniformly high order accurate discontinuous Galerkin (DG) and weighted essentially non-oscillatory (WENO) finite volume (FV) schemes satisfying a strict maximum principle for scalar conservation laws and passive convection in incompressible flows, and positivity preserving for density and pressure for compressible Euler equations. A general framework (for arbitrary order of accuracy) is established to construct a limiter for the DG or FV method with first order Euler forward time discretization solving one dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle and make the scheme uniformly high order in space and time. One remarkable property of this approach is that it is straightforward to extend the method to two and higher dimensions. The same limiter can be shown to preserve the maximum principle for the DG or FV scheme solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. A suitable generalization results in a high order DG or FV scheme satisfying positivity preserving property for density and pressure for compressible Euler equations. Numerical tests demonstrating the good performance of the scheme will be reported. This is a joint work with Xiangxiong Zhang.

Consider the following situation. We are given an element in a Banach algebra and the following operations are assumed: one can form polynomials from the algebra element and one can ask for the norm of such a polynomial. Also, we are given an analytic function of which we can ask values of it and its derivatives at points of our choice. Can we have an algorithm which evaluates the analytic function at the Banach algebra element, whenever it can be consistently defined. One needs to know the spectrum of the element and then the function can be evaluated provided the function is analytic in a neighborhood of the convex hull of the spectrum. We have recently shown [1] that there exists an algorithm to represent the resolvent outside any open neighborhood of the polynomially convex hull of the spectrum. Then the holomorphic functional calculus can be used to define the function at the element. In this talk we focus on computational aspects of using the representation of the resolvent within the Cauchy integral. The resolvent is written explicitly as a power series with help of just one polynomial. This in a natural way leads to a multicentric power series (Jacobi series) for the holomorphic function. The formulas are spelled out in [2].
References.
[1] O.Nevanlinna, Numer Funct Anal Optim 30, 1025-1047, 2009
[2] O.Nevanlinna, Multicentric holomorphic calculus, Helsinki University of Technology, Mathematics Research Reports, A596, 2010

Krylov solvers are currently the standard tool for solving very large sparse linear systems of equations by iteration. But for most real-world problems they only converge in a reasonable number of iterations if a suitable preconditioning technique is applied. This is particularly true for problems where the linear operator has eigenvalues of small absolute value | a situation that is very common in practice. One suitable technique for dealing with such problems is to identify an approximately invariant subspace $\mathcal Z$ that belongs to the set of these small eigenvalues. By using an orthogonal projection along $\mathcal Z$ the Krylov solver can then be applied only to the orthogonal complement by restricting the operator accordingly. The basis constructed implicitly or explicitly by this restricted operator should then be augmented by a set of basis vectors for $\mathcal Z$. There are various ways to handle and implement this approach. They differ not only algorithmically and numerically, but sometimes also mathematically. Some keywords associated with such methods are `(spectral) deflation', `augmented basis', `recycling Krylov subspaces', and `singular preconditioning'.

While we quickly also review the `symmetric case', where the linear system is Hermitian (or real and symmetric), we are mostly interested in the `non-symmetric case', where our main message is that the orthogonal projection should be replaced by a suitable oblique projection, so that when its nullspace is invariant, so is its range.

We shall experience self-calibrating computational models with applications in both geo- and life sciences. The automated implementation in the FEniCS software suite will be illustrated.

In order to investigate turbulent phenomena in compressible flows the Variational Multiscale Method (VMS) is used. This method is usually applied to incompressible flows. In a first step, we derive the VMS method for the compressible Navier-Stokes equations. The resulting weak formulation of the flow equations is split into resolved and unresolved scales using multiresolution techniques based on biorthogonal wavelets. Since the in principle infinite-dimensional subspace of fluctuations is also discretized, it needs to be stabilized by additional dissipative terms. The compressible VMS method is then incorporated into an adaptive multiresolution finite volume solver, where grid adaptation is also performed by means of the multiresolution analysis. Finally, first results are presented for a turbulent channel flow. This is joint work with Wolfgang Dahmen, Thomas Gotzen and Roland Schäfer.

The residual distribution schemes (RD), or fluctuation splitting schemes, have been introduced by P.L. Roe in several papers, among which his 1981 JCP paper, and then in collaboration with VKI. Considering a transport equation, the idea is to represent the solution by its nodal values, the mesh being described in a non structured manner, and to evolve it using the most possible compact stencil with the highest possible accuracy that can be reached with the scheme stencil without sacrificing the non oscilalatory properties of the exact solutions. For example, using a triangular mesh, one can get second order accuracy only using the nearest neighbors of any given vertex. The discrete scheme formulation shares many similarities with stabilised finite element, though they are very different in that the schemes can be proved to be L ∞ stable. They also share many similarities with standard finite volume schemes since any finite volume scheme can be reformulated as a RD scheme, at least in their first order version. Of course the classical MUSCL schemes violate the spencil compactness requirements of RD schemes, hence at least from a pure computational point of view, the second order version of the RD schemes are superior to standard second order FV scheme. Since the early versions of RD scheme, many improvement have been realised. First, it is possible to construct very high order schemes on hybrid meshes in two and three dimensions. Efficient unsteady schemes exists now. Recently, it has been possible to design very high order non oscilaltory schemes for advection diffusion equations. In this lecture, we will describe the design principle of RD scheme and show how very high order can be reached. Efficient unsteady version will be presented as well as the very high order approximation of viscous problems. I will also present some open questions.

In this talk, we will present moving grid strategies for solving hyperbolic conservations laws. In particular, we will discuss how to combine the Discontinuous Galerkin Method and the moving mesh methods.
Some relevant references can be found in http://www.math.hkbu.edu.hk/~ttang/moving.html

Abstract. We introduce a new particle-based model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The new model does not involve any explicit dependence on the number of agents: only their self-driven geometry in phase space enters the model. It lacks, however, the symmetry property, which is the key y for the various studies of C-S flocking behavior. To this end, we introduce here a new unifying framework to analyze the phenomenon of flocking for a general class of dynamical systems in the presence of non-symmetric influence matrices. In particular, we prove the emerging behavior of flocking in the proposed model, when the pairwise long-range interactions between its agents decays sufficiently slow. The methodology presented in this paper is based on the notion of active sets, which carries over from the particle to the kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic description of our new model for self-organized dynamics, and we prove its unconditional flocking for sufficiently slowly decaying influence functions.