Eduard Feireisl (Academy of Sciences, Prague, Czech Republic)
Singular limits in thermodynamics of viscous fluids (Abstract)

Stig Larsson (Chalmers University, Göteborg, Sweden)
Numerical solution of stochastic partial differential equations driven by noise (Abstract)

Mechthild Thalhammer (University of Innsbruck, Austria)
Time-splitting spectral methods for nonlinear Schrödinger equations (Abstract) (Lecture notes)

Julien Vovelle (University of Lyon, France)
Stochastic perturbation of scalar conservation laws (Abstract) (Lecture notes)



Eduard Feireisl: Singular limits in thermodynamics of viscous fluids

Singular limits in equations and systems of fluid mechanics is a topic of considerable theoretical interest with numerous applications in numerical analysis as well as practical experiments. In this series of lectures, we discuss the singular limits from the perspective of the theory of weak solutions for both the primitive and the target system. A dimensionless variant of the standard Navier-Stokes-Fourier system contains a family of characteristic numbers that may be either small or large as the case may be. Our goal is to discuss the behaviour of solutions to this (primitive) system provided certain of these parameters become singular, in particular, we identify the limit (target) problem. To this end, the following issues will be addressed:

The abstract theoretical results will be illustrated by several examples including: (i) the incompressible limit in the low Mach number regime, where the target problem is the Oberbeck-Boussinesq approximation, (ii) the incompressible limit for low Mach and Froude numbers, where we obtain the so-called anelastic approximation including several practical examples of models in astrophysics, (iii) the acoustic equation proposed by Lighthill and its analysis by means of the methods of two-scale convergence.

Stig Larsson: Numerical solution of stochastic partial differential equations driven by noise

We study evolution partial differential equations driven by noise, for example, the stochastic heat equation or the stochastic wave equation, where the source term (right-hand side) is supplemented by a noise term. The noise can be uncorrelated in space and time (white noise) or spatially correlated.
We begin by briefly presenting an abstract framework in which such equations can be given a rigorous meaning. The framework is based on the theory of semigroups of bounded linear operators in Hilbert space. The equations are discretised by a standard finite element method in the spatial variable and by the Euler method in the temporal variable. The discrete equations are set in the same abstract framework. We show convergence estimates. We also discuss how the numerical methods can be implemented in a computer program. Finally, we review other approaches to error analysis that have appeared in the literature.

  1. Introduction. Basics of semigroup theory.
  2. The stochastic heat and wave equations.
  3. Finite element approximation. Error estimates for the deterministic equations. Strong convergence estimates for the stochastic equations.
  4. Review of other approaches to error analysis.

Mechthild Thalhammer: Time-splitting spectral methods for nonlinear Schrödinger equations

In this lecture, we are concerned with efficient numerical methods for the time integration of nonlinear Schrödinger equations. As a model problem, we consider the Gross-Pitaevskii equation describing the quantum physical phenomenon of Bose-Einstein condensation. Our intention is to study the quantitative and qualitative behaviour of high-accuracy discretisations that rely on time-splitting Hermite and Fourier pseudo-spectral methods. In particular, this includes a stability and convergence analysis of high-order exponential operator splitting methods for evolutionary Schrödinger equations. Numerical examples for the Gross-Pitaevskii equation illustrate the theoretical results.

Julien Vovelle: Stochastic perturbation of scalar conservation laws

We give a brief introduction to the well-posedness theory and kinetic formulation of first-order scalar conservation laws and to the numerical approximation by the finite volume method. The second part of the course will be concerned with the long-time behaviour of solutions to first-order scalar conservation laws. In the third part, we shall study invariant measures for the stochastic perturbation of conservation laws. If time permits, a fourth part will address numerical issues in relation with the approximation and behaviour of the randomly forced equation.