Talks

This presentation introduces the semi-implicit Smoothed Particle Hydrodynamics (SPH)
scheme [1] for the shallow water equations following the semi-implicit finite volume and finite
difference approach of Casulli [2]. In standard explicit numerical methods, there is often a severe
limitation on the time step due to the stability restriction imposed by the CFL condition. To this
effect, a semi-implicit SPH scheme is derived, which leads to an unconditionally stable method.
The discrete momentum equation is substituted into the discrete continuity equation to obtain
a symmetric positive definite linear system for the free surface elevation. The resulting system
can be easily solved by a matrix-free conjugate gradient method. Once the new free surface
location is known, the velocity at the new time level can be directly computed and the particle
positions can subsequently be updated. We further discuss a nonlinear algorithm for treating
wetting/drying problems. We derive a mildly nonlinear system for the discrete free surface
elevation from the shallow water equations by taking into consideration a correct mass balance
in wet regions and in transition regions, i.e. the regions from wet particles to dry particles
and those from dry particles to wet particles. The scheme is validated on a two dimensional
inviscid hydrostatic free surface flows for the two dimensional shallow water equations and
wetting/drying test problem.

References
[1] A.O. Bankole, A. Iske, T. Rung, M. Dumbser, A meshfree semi-implicit Smoothed Particle
Hydrodynamics method for free surface flow. Meshfree Methods for Partial Differential
Equations VIII, M. Griebel and M.A. Schweitzer (eds.), Springer LNCSE, Vol. 115, pp.
35-52 (2017).
[2] V. Casulli, Semi-Implicit Finite Difference Methods for the Two-Dimensional Shallow
Water Equations. Jour. of Comp. Phys., Vol 86. pp. 56-74 (1990).

Meeting-ID: 820 3979 6993
Passwort: 694649

Solving differential equations rigorously is a main and vigorous topic in the
field of verified computation. Here, solving rigorously means that a computer
program supplies an approximate solution along with error bounds that respect
all numerical as well as all rounding errors that occurred during the computation.
An exact solution is proved to be enclosed within these rigorous bounds.
In this context so-called Taylor models have been used successfully for solving
ordinary differential equations (ODEs) rigorously. Implementations are COSY INFINITY [1], FLOW [2], ODEIntegretor [3], and RIOT [4]. Here, COSY INFINITY
developed by Berz and Makino and their group is the most advanced
implementation. Recently, we implemented the Taylor model approach in MATLAB/
INTLAB [5].

We give a short introduction to Taylor models, their rigorous arithmetic,
and the Taylor model method for enclosing solutions of ordinary differential
equations in a verified manner. We only treat initial value problems
$y_0 = f(t,y)$, $y(t_0) = y_0$
where the initial value $y_0$ may be an interval vector. For specific ODEs we demonstrate
how to use and call our verified ODE solver. This is designed to be very
similar to calling MATLAB's non-verified ODE solvers like ode45. Finally, results
and run times are compared to those of COSY INFINITY, RIOT and Lohner's
classical AWA.

[1] M. Berz, K. Makino, COSY INFINITY: www.bt.pa.msu.edu/index_cosy.htm
[2] X. Chen, Reachability analysis of non-linear hybrid systems using Taylor models,
Dissertation RWTH Aachen, 2015. FLOW: https://flowstar.org/dowloads/
[3] T. Dzetkulic, Rigorous integration of non-linear ordinary differential equations in
Chebyshev basis, Numer. Algor. 69, 183-205, 2015.
ODEintegrator: https://sourceforge.net/projects/odeintegrator
[4] I. Eble, Über Taylor-Modelle, Dissertation at Karlsruhe Inst. of Technology, 2007.
RIOT: www.math.kit.edu/ianm1/~ingo.eble/de
[5] S.M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing
(ed. by Tibor Csendes), Kluwer Academic Publishers, 77-104, 1999.
INTLAB: http://www.ti3.tu-harburg.de/intlab/

Talk (PDF, 100KB)

https://tuhh.zoom.us/j/87535538628?pwd=RTJ0ZGp0ZWc1NVk3RGp5NTBQYjhVdz09

Meeting ID: 875 3553 8628
Passcode: 750232

Production processes are usually investigated using models and methods from queueing theory (queue = line where people wait for goods or services). Control of warehouses and their optimization rely on models and methods from inventory theory. Both theories are fields of Operations Research, but they comprise quite different methodologies and techniques. In classical Operations Research these theories are considered as disjoint research areas. Today's emergence of complex supply chains (=production-inventory networks) calls for integrated production-inventory models, which are focus of my research. We have developed Markov process models for several production-inventory systems and derived the steady state distribution of the global system. For most of the production-inventory systems the obtained steady state is of product form. This enables us to analyse the long term average costs with the aim to find the optimal inventory size.
In my talk, I focus on a basic production-inventory model and present the essentials of the other models. Furthermore, I show the connection to the industrial project “Robotic Mobile fulfillment system”.

*title by Karsten Kruse