Fakultät für Mathematik und Naturwissenschaften

Artificial Boundary Conditions

When computing numerically the solution of a partial differential equation in an unbounded domain usually artificial boundaries are introduced to limit the computational domain. Special boundary conditions are derived at this artificial boundaries to approximate the exact whole-space solution. If the solution of the problem on the bounded domain is equal to the whole-space solution (restricted to the computational domain) these boundary conditions are called transparent boundary conditions (TBCs).

We are concerned with TBCs for general Schrödinger-type pseudo-differential equations arising from `parabolic' equation (PE) models which have been widely used for one-way wave propagation problems in various application areas, e.g. (underwater) acoustics, seismology, optics and plasma physics. As a special case the Schrödinger equation of quantum mechanics is included.

Existing discretizations of these TBCs induce numerical reflections at this artificial boundary and also may destroy the stability of the used finite difference method. These problems do not occur when using a so-called discrete TBC which is derived from the fully discretized whole-space problem. This discrete TBC is reflection-free and conserves the stability properties of the whole-space scheme. We point out that the superiority of discrete TBCs over other discretizations of TBCs is not restricted to the presented special types of partial differential equations or to our particular interior discretization scheme.

Another problem is the high numerical effort. Since the discrete TBC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for long-time simulations. As a remedy we construct new approximative TBCs involving exponential sums as an approximation to the convolution kernel. This special approximation enables us to use a fast evaluation of the convolution type boundary condition.

Finally, to illustrate the broad range of applicability of our approach we derived efficient discrete artificial boundary conditions for the Black-Scholes equation of American options.

Software

Our approach was implemented by C.A. Moyer in the QMTools software package for quantum mechanical applications.

Publications



2022

5629.

Bohrmann-Linde, Claudia; Siehr, Ilona
CHEMIE Einführungsphase Nordrhein-Westfalen
Herausgeber: C.C.Buchner Verlag, Bamberg
August 2022

ISBN: 9783661060019

5628.

[german] Monique, Meier; Zeller, Diana; Stinken-Rösner, Lisa
Interaktive Videoformate für den naturwissenschaftlichen Unterricht. Vom Rezipieren zum Interagieren
Unterricht Biologie, 475 :44-47
07 2022

5627.

[german] Grandrath, Rebecca; Bohrmann-Linde, Claudia
Strom aus Bäckerhefe
Nachrichten aus der Chemie, 70 (7-8) :18-21
Juli 2022

5626.

[german] Zeller, Diana
Medialab – ein dreistufiges Modul zur Entwicklung digitalisierungsbezogener Kompetenzen im Studium des Chemie‐ und Sachunterrichtslehramts
CHEMKON, 29 (S1) :287-292
Juni 2022

5625.

[german] Grandrath, Rebecca; Bohrmann-Linde, Claudia
Entwicklung eines lowcost Experiments für den Chemieunterricht am Beispiel der enzymatischen Brennstoffzelle mit Lactase
CHEMKON, 29 (S1) :233-238
Juni 2022

5624.

[english] Bohrmann-Linde, Claudia; Zeller, Diana; Meuter, Nico; Tausch, Michael W.
Teaching Photochemistry: Experimental Approaches and Digital Media
ChemPhotoChem, 6 (6) :1-11
Juni 2022

5623.

[german] Zeller, Diana; Meier, Monique
Videos interaktiv erweitern - Forschendes Lernen vielseitig unterstützen
Digital Unterricht Biologie, 4 :10-11
Mai 2022

5622.

[german] Gökkus, Yasemin; Tausch, Michael W.
Explorative Studie zur partizipativen und nutzenorientierten Forschung in der Chemiedidaktik
CHEMKON, 29 (3) :117-124
April 2022

5621.

Frommer, Andreas; Kahl, Karsten; Schweitzer, Marcel; Tsolakis, Manuel
Krylov subspace restarting for matrix Laplace transforms
2022

5620.

Gerlach, Moritz; Glück, Jochen
On characteristics of the range of integral operators
2022

5619.

Bolten, M.; De Sturler, E.; Hahn, C.
Krylov Subspace Recycling for Evolving Structures
Comput. Methods Appl. Mech. Engrg., 391 :114222
2022

5618.

Klamroth, Kathrin; Stiglmayr, Michael; Sudhoff, Julia
Ordinal Optimization Through Multi-objective Reformulation
math.OC, arXiv:2204.02003
2022
Herausgeber: arXiv

5617.

Jacob, Birgit; Morris, Kirsten
On solvability of dissipative partial differential-algebraic equations
IEEE Control. Syst. Lett., 6 :3188-3193
2022

5616.

Farkas, Bálint; Jacob, Birgit; Schmitz, Merlin
On exponential splitting methods for semilinear abstract Cauchy problems
2022

5615.

Botchev, M. A.; Knizhnerman, L. A.; Schweitzer, M.
Krylov subspace residual and restarting for certain second order differential equations
2022

5614.

Klamroth, Kathrin; Stiglmayr, Michael; Sudhoff, Julia
Multi-objective Matroid Optimization with Ordinal Weights
Discrete Applied Mathematics
2022

5613.

Frommer, Andreas; Kahl, Karsten; Schweitzer, Marcel; Tsolakis, Manuel
Krylov subspace restarting for matrix Laplace transforms
2022

5612.

Botchev, M. A.; Knizhnerman, L. A.; Schweitzer, M.
Krylov subspace residual and restarting for certain second order differential equations
2022

5611.

Daners, Daniel; Glück, Jochen; Mui, Jonathan
Local uniform convergence and eventual positivity of solutions to biharmonic heat equations
2022

5610.

[german] Tausch, Michael W.
LED statt Gasbrenner - Mehr Licht für nachhaltigen Chemieunterricht
Chemie in unserer Zeit, 56 (3/2022) :188–196
2022

5609.

Frommer, Andreas; Kahl, Karsten; Schweitzer, Marcel; Tsolakis, Manuel
Krylov subspace restarting for matrix Laplace transforms
2022

5608.

[german] Banerji, Amitabh; Dörschelln, Jennifer; Schwarz, D.
Organische Leuchtdioden im Chemieunterricht
Chemie in unserer Zeit, 52 (1) :34-41
2022

5607.

[german] Zeller, Diana; Bohrmann-Linde, Claudia
#debunk YouTube-Videos - Ein didaktisches Konzept zum Einsatz von Videos im Chemieunterricht zur Stärkung der Digital Scientific Literacy
MNU journal, 75 (03) :197-201
2022

5606.

Ballaschk, Frederic; Kirsch, Stefan F.
Oxidations with Iodine(V) Compounds – From Stoichiometric Compounds to Catalysts
In Ishihara, Kazuaki and Muñiz, Kilian, Editor, Iodine Catalysis in Organic Synthesis
Seite 299–334
Herausgeber: Wiley
1 Edition
2022
299–334

5605.

Teng, L.; Ehrhardt, M.; Günther, M.
Stochastic Correlation: Modelling, Analysis and Numerical Simulation with Applications in Finance
Herausgeber: World Scientific

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