TY  - JOUR
A1  - Breuß, Michael
A1  - Kleefeld, Andreas
T1  - Implicit monotone difference methods for scalar conservation laws with source terms
JF  - Acta Mathematica Vietnamica
N2  - In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970–986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.
KW  - Entropy solution
KW  - Source term
KW  - Monotone methods
KW  - Implicit methods
KW  - Finite difference methods
KW  - Conservation laws
Y1  - 2020
U6  - https://doi.org/10.1007/s40306-019-00354-1
SN  - 2315-4144
N1  - Corresponding author: Andreas Kleefeld
VL  - 45
SP  - 709
EP  - 738
PB  - Springer Singapore
CY  - Singapore
ER  - 
TY  - CHAP
A1  - Kahra, Marvin
A1  - Breuß, Michael
A1  - Kleefeld, Andreas
A1  - Welk, Martin
ED  - Brunetti, Sara
ED  - Frosini, Andrea
ED  - Rinaldi, Simone
T1  - An Approach to Colour Morphological Supremum Formation Using the LogSumExp Approximation
T2  - Discrete Geometry and Mathematical Morphology
N2  - Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations.
In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric 2x2 matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.
Y1  - 2024
SN  - 978-3-031-57793-2
U6  - https://doi.org/10.1007/978-3-031-57793-2_25
N1  - Third International Joint Conference, DGMM 2024, Florence, Italy, April 15–18, 2024
SP  - 325
EP  - 337
PB  - Springer
CY  - Cham
ER  -