Lectures
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)
Abstracts
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:
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.
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.