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
- 2025
5407.
Solid-Supported Iodine(V) Reagents in Organic Synthesis
Chemistry - A European Journal, 2025 :e202500670
03 2025
Herausgeber: Wiley
ISSN: 1521-37655406.
[German] Grandrath, Rebecca; Wiebel, Michelle; Bensberg, Kathrin; Schebb, Nils Helge; Bohrmann-Linde, Claudia
Aus der Schale in die Schule
Nachrichten aus der Chemie, 73 (3) :10-12
März 20255405.
Aus der Schale in die Schule
Nachrichten aus der Chemie, 2025 :10-12
02 2025
Herausgeber: Wiley
ISSN: 1868-00545404.
Asya, Berçin V.; Wang, Sitao; Euchler, Eric; Khiêm, Vu Ngoc; Göstl, Robert
Optical Force Probes for Spatially Resolved Imaging of Polymer Damage and Failure
Aggregate, 6 :e70014
Februar 2025
ISSN: 2692-45605403.
Hahmann, Johannes; Schüpp, Boris N.; Ishaqat, Aman; Selvakumar, Arjuna; Göstl, Robert; Gräter, Frauke; Herrmann, Andreas
Sequence-specific, mechanophore-free mechanochemistry of DNA
Chem, 11 :102376
Januar 2025
ISSN: 2451-9294, 2451-93085402.
Clevenhaus, A.; Totzeck, C.; Ehrhardt, M.
A Space Mapping approach for the calibration of financial models with the application to the Heston model
20255401.
Frommer, Andreas; Rinelli, Michele; Schweitzer, Marcel
Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices
Math. Comp., 94 :801-823
20255400.
Hoffe, Leon; Ulutas, Berna; Klamroth, Kathrin; Bracke, Stefan
Assessing the effectiveness and efficiency of selected solution approaches for two-dimensional stock cutting problems (Part III): Hybrid Approach For Printed Circuit Boards
AUTOMATION 2025: Conference on Automation — Innovations and Future Perspectives
20255399.
Kiesling, Elisabeth; Bohrmann-Linde, Claudia
Carbon Capture and Storage - Nachweis von adsorbiertem Kohlenstoffdioxid
Naturwissenschaften im Unterricht Chemie, 1/25 :Versuchskarteikarte
20255398.
Clément, François; Doerr, Carola; Klamroth, Kathrin; Paquete, Luís
Constructing Optimal Star Discrepancy Sets
accepted in Proceedings of the AMS
20255397.
Kunze, Markus; Mui, Jonathan; Ploss, David
Elliptic operators with non-local Wentzell-Robin boundary conditions
20255396.
Song, Yongcun; Wang, Ziqi; Zuazua, Enrique
FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning
Neural Network, 181
Januar 20255395.
Kienitz, J; Moodliyar, L
Gaussian views explained
Wilmott, 2025 (135) :72–77
2025
Herausgeber: Wilmott Magazine5394.
Xu, Zhuo; Tucsnak, Marius
Global Exponential Stabilization for a Simplified Fluid-Particle Interaction System
Januar 20255393.
Bartel, Andreas; Schaller, Manuel
Goal-oriented time adaptivity for port-Hamiltonian systems
Journal of Computational and Applied Mathematics, 461 :116450
2025
ISSN: 0377-04275392.
Schäfers, Kevin; Finkenrath, Jacob; Günther, Michael; Knechtli, Francesco
Hessian-free force-gradient integrators
Computer Physics Communications, 309 :109478
2025
ISSN: 0010-46555391.
Schäfers, Kevin; Finkenrath, Jacob; Günther, Michael; Knechtli, Francesco
Hessian-free force-gradient integrators
Computer Physics Communications, 309 :109478
2025
ISSN: 0010-46555390.
Schäfers, Kevin; Finkenrath, Jacob; Günther, Michael; Knechtli, Francesco
Hessian-free force-gradient integrators and their application to lattice QCD simulations
PoS, LATTICE2024 :025
20255389.
Schäfers, Kevin; Finkenrath, Jacob; Günther, Michael; Knechtli, Francesco
Hessian-free force-gradient integrators and their application to lattice QCD simulations
PoS, LATTICE2024 :025
20255388.
Storch, Sonja; Campagna, Davide; Aydonat, Simay; Göstl, Robert
Mechanochemical generation of nitrogen-centred radicals for the formation of tertiary amines in polymers
RSC Mechanochemistry, 2
Januar 20255387.
Schweitzer, Marcel
Near instance optimality of the Lanczos method for Stieltjes and related matrix functions
20255386.
Bolten, Matthias; Doganay, Onur Tanil; Gottschalk, Hanno; Klamroth, Kathrin
Non-convex shape optimization by dissipative Hamiltonian flows
Engineering Optimization, 57 :384--403
20255385.
Beck, Christian; Jentzen, Arnulf; Kleinberg, Konrad; Kruse, Thomas
Nonlinear Monte Carlo Methods with Polynomial Runtime for Bellman Equations of Discrete Time High-Dimensional Stochastic Optimal Control Problems
Appl. Math. Optim., 91 (1) :26
20255384.
Lorenz, Jan; Zwerschke, Tom; Günther, Michael; Schäfers, Kevin
Operator splitting for coupled linear port-Hamiltonian systems
Applied Mathematics Letters, 160 :109309
2025
Herausgeber: Elsevier5383.
Lorenz, Jan; Zwerschke, Tom; Günther, Michael; Schäfers, Kevin
Operator splitting for coupled linear port-Hamiltonian systems
Applied Mathematics Letters, 160 :109309
2025
Herausgeber: Elsevier