Talks

The behaviour of particulate flow is mathematically modelled by population balance equations. The various terms of the equation model phenomena including particle transport, nucleation, growth, and aggregation. Their efficient numerical simulation requires sophisticated techniques, and various approaches proposed in the literature vary not only in computational complexity but also in the accuracy of the computed solutions. We will focus on the numerical treatment of aggregation integrals, the terms that model the aggregation process and which oftentimes dominate the overall simulation time. Within such a process, particles are characterized by a property coordinate x, e.g. the particle mass, the particle area, or the chemical composition, to mention only a few, and their distribution is quantified by a density distribution function f(x,t), which describes the property distribution of the particles at a given time t.

First, we discuss the evaluation of univariate aggregation integrals, where only one of particle characteristics is considered, and we discretise the property coordinate x through equidistant grids and approximate the density distribution f(x,t) through piecewise constant functions. Then, we extend the approach to grids with nested structures and approximation the density distribution through higher order polynomials (of degree p), which allow a better approximation. This novel approach reduces the quadratic complexity of its direct computation to an almost optimal complexity of order pNlogN with the problem size N. Furthermore, we also discuss examples of bivariate problems, where also a second property of particles is considered.

The key components of the developed algorithms are a separable approximation of the aggregation kernel, a nested grid consisting of piecewise uniform portions, application of FFT to compute the aggregation (convolution) on such uniform portions and orthogonality of basis functions which in combination lead to efficient recursion formulas. We provide extensive numerical tests for different initial setups to illustrate the performance of the developed algorithms with respect to their accuracy and efficiency, leading to (heuristic) strategies for the choice of discretization parameters.

Talk (PDF, 228KB)

Scattered data interpolation refers to an interpolation problem where the data sites are distributed irregularly within some domain. An interpolant may be constructed as a linear combination of radial basis functions centered at the data sites. Finding the coefficients in this representation leads to linear equations where the system matrices are large, dense, indefinite and ill-conditioned. These matrices can be approximated using the framework of hierarchical matrices. We will compare different approximation methods and discuss how to construct algebraic preconditioners.

Es werden große dünnbesetzte Sattelpunktprobleme betrachtet, wie sie z.B. in der Strömungsmechanik auftreten. Diese i.A. unsymmetrischen und indefiniten Systeme können mittels iterativer Krylovraum-Verfahren, inkl. geeigneter Präkonditionierer, gelöst werden. Insbesondere werden sogenannte induzierte Dimensions-Reduktions-Methoden (IDR), im Speziellen QMRIDR(s), verwendet, welche zusätzlich mit einem Deflationsansatz gepaart werden. Dabei werden Informationen aus früheren Durchläufen derart recycelt, sodass es möglich ist, Sequenzen von linearen Systemen effektiv zu lösen. Als Beispiel werden die diskretisierten Oseen-Gleichungen betrachtet; weitere Anwendung kann dies darüber hinaus z.B. bei inneren Punkte-Verfahren in der linearen Optimierung finden.

The goal of an a posteriori error analysis for an approximate PDE
solution is to establish the equivalence of error and a posteriori
estimator. Unfortunately, this equivalence is often only up to so-
called oscillation terms.

In this talk we shall clarify the reasons for the presence of
oscillation. Moreover, we propose a new approach to a posteriori error
estimation, where oscillation can be bounded by the error and so does
not longer spoil the aforementioned equivalence.

This is joint work with Christian Kreuzer (Bochum).

We present a space-time discontinuous Galerkin (DG) method for linear
wave propagation problems.
The special feature of the scheme is that it is a Trefftz method,
namely that trial and test functions are solution of the partial
differential equation to be discretised in each element of the
(space-time) mesh.
The DG scheme is defined for unstructured meshes whose internal faces
need not be aligned to the space-time axes.
The Trefftz approach can be used to improve and ease the
implementation of explicit schemes based on ``tent-pitched'' meshes.
We show that the scheme is well-posed, quasi-optimal and dissipative,
and prove a priori error bounds for general Trefftz discrete spaces.
A concrete discretisation can be obtained using piecewise polynomials
that satisfy the wave equation elementwise, for which we show high
orders of convergence.
If time allows, we will describe a similar Trefftz-DG method for the
Helmholtz equation, i.e. wave equation in time-harmonic regime, for
which non-polynomial basis functions are used and quite a complete
theory has been established.