Applied and Computational Mathematics (ACM)

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



2024

6855.

Bauß, Julius; Stiglmayr, Michael
Adapting Branching and Queuing for Multi-objective Branch and Bound
Operations Research Proceedings 2023
Herausgeber: Springer
2024
2023

6854.

[german] Kiesling, Elisabeth; Kremer, Richard; Pereira Vaz, Nuno; Venzlaff, Julian; Bohrmann-Linde, Claudia
Wege aus der Klimakrise – ein BNE-Schülerlaborangebot mit mehrdimensionalem Zugang
MNU Journal, 76 (06/2023)
November 2023
ISSN: 0025-5866

6853.

[english] Venzlaff, Julian; Kosumi, Kaltrina; Zeller, Diana; Bohrmann-Linde, Claudia
Education for Sustainable Development and Experiments involving Titanium Dioxide
World Journal of Chemical Education, 11 (3) :80-86
August 2023

6852.

[english] Brunnert, Rainer; Tausch, Michael W.
Green Chemistry in STEM Education: Light for Basic Concepts
World Journal of Chemical Education, 11 (3) :65-73
August 2023

6851.

[german] Tausch, Michael W.; Schneidewind, Jacob
Mit Licht zu grünem Wasserstoff
Chemie in unserer Zeit
August 2023

6850.

Clevenhaus, A.; Totzeck, C.; Ehrhardt, M.
A numerical study of the impact of variance boundary conditions for the Heston model
IMACM preprint 23/11
Juli 2023

6849.

[german] Venzlaff, Julian; Bohrmann-Linde, Claudia
Photoreformierung – grüner Wasserstoff aus Alkohol, Zucker und Stärke
CHEMKON, 30 (6) :254-259
Juni 2023

6848.

Mittendorf, Fabia; Quambusch, Moritz; Kirsch, S. F.
Total synthesis of both enantiomers of the biosurfactant aureosurfactin via bidirectional synthesis with a chiral Horner–Wittig building block
Organic & Biomolecular Chemistry
05 2023

6847.

[german] Grandrath, Rebecca; Bohrmann-Linde, Claudia
Dem Apfel ans Leder
Nachrichten aus der Chemie, 71 :12-14
März 2023

6846.

Ehrhardt, Matthias; Zheng, Chunxiong
für Angewandte Analysis und Stochastik

6845.

Ehrhardt, Matthias
Ein einfaches Kompartment-Modell zur Beschreibung von Revolutionen am Beispiel des Arabischen Frühlings

6844.

Günther, Michael
Einführung in die Finanzmathematik

6843.

Gjonaj, Erion; Bahls, Christian Rüdiger; Bandlow, Bastian; Bartel, Andreas; Baumanns, Sascha; Belzen, F; Benderskaya, Galina; Benner, Peter; Beurden, MC; Blaszczyk, Andreas; others
Feldmann, Uwe, 143 Feng, Lihong, 515 De Gersem, Herbert, 341 Gim, Sebasti{\'a}n, 45, 333
MATHEMATICS IN INDUSTRY 14 :587

6842.

Ehrhardt, Matthias; Zheng, Chunxiong
für Angewandte Analysis und Stochastik

6841.

Hendricks, C; Ehrhardt, M; Günther, M
High order tensor product interpolation in the Combination Technique
preprint, 14 :25

6840.

Ehrhardt, Matthias; Günther, Michael; Striebel, Michael
Geometric Numerical Integration Structure-Preserving Algorithms for Lattice QCD Simulations

6839.

Tripiccione, Betreuer Raffaele; Ehrhardt, Matthias; Alexandrou, Constantia; Toschi, Federico; Simma, Hubert; Schifano, Co-Betreuer Sebastiano Fabio
Daniele Simeoni 1836010

6838.

Hendricks, C; Ehrhardt, M; Günther, M
High order tensor product interpolation in the Combination Technique
preprint, 14 :25

6837.

Hendricks, Christian; Ehrhardt, Matthias; Günther, Michael
Hybrid finite difference/pseudospectral methods for stochastic volatility models
19th European Conference on Mathematics for Industry, Seite 388

6836.

Hendricks, Christian; Ehrhardt, Matthias; Günther, Michael
Hybrid finite difference/pseudospectral methods for stochastic volatility models
19th European Conference on Mathematics for Industry, Seite 388

6835.

Ambartsumyan, I; Khattatov, E; Yotov, I; Zunino, P; Arnold, Anton; Ehrhardt, Matthias; Ashyralyev, Allaberen; Csom{\'o}s, Petra; Farag{\'o}, Istv{\'a}n; Fekete, Imre; others
Invited Papers

6834.

Ambartsumyan, I; Khattatov, E; Yotov, I; Zunino, P; Arnold, Anton; Ehrhardt, Matthias; Ashyralyev, Allaberen; Csomós, Petra; Faragó, István; Fekete, Imre; others
Invited Papers

6833.

Günther, Michael; Kossaczk{\`y}, Igor
Lab Exercises for Numerical Analysis and Simulation I: ODEs

6832.

Ehrhardt, Matthias; Günther, Michael; Brunner, H; Dalhoff, A
Mathematical Modelling of Dengue Fever Epidemics

6831.

Ehrhardt, Matthias
Ein einfaches Kompartment-Modell zur Beschreibung von Revolutionen am Beispiel des Arabischen Frühlings

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