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Arbeitsgruppe Angewandte Mathematik / Numerische Analysis
Bergische Universität Wuppertal
Fakultät 04
Gaußstraße 20
D-42119 Wuppertal
Deutschland

Telefon: +49 202 439 5296
Fax: +49 202 439 5201
E-Mail: sek-amna{at}math.uni-wuppertal.de

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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

Referenzen
44.
P. Putek; E. J. W. ter Maten; M. Günther; A. Bartel; R. Pulch; P. Meuris; W. Schoenmaker
Robust Shape Optimization under Uncertainties in Device Materials, Geometry and Boundary Conditions
Mathematics in Industry
Seite 223--260.
Herausgeber: Springer International Publishing,
2019
43.
H. Espinoza; R. Paredes; M. Ehrhardt
A New Two-Way Artificial Boundary Condition for Wave Propagation: Formulation and Benchmarking
23rd AIAA/CEAS Aeroacoustics Conference, AIAA Aviation and Aeronautics Forum and Exposition, Colorado, 5-9 June 2017
2017
42.
C. Besse; M. Ehrhardt; I. Lacroix-Violet
Discrete Artificial Boundary Conditions for the Korteweg de Vries equation
Numer. Meth. Part. Diff. Eqs., 32(5):1455--1484
2016
41.
P. Petrov; M. Ehrhardt
Transparent boundary conditions for iterative high-order parabolic equations
J. Comput. Phys., 313:144--158
2016
40.
D. Heubes; A. Bartel; M. Ehrhardt
Concept for a one-dimensional discrete artificial boundary condition for the lattice Boltzmann method
Comput. Math. Appl., 70(10):2316--2330
2015
39.
P. Petrov; M. Ehrhardt
On Mayfield's stability proof for the discretized transparent boundary condition for the parabolic equation
Appl. Math. Lett., 44:45--49
2015
38.
P. Petrov; M. Ehrhardt
Transparent boundary conditions for the high-order parabolic approximations
Proceedings of the International Conference DAYS on DIFFRACTION 2015, St.Petersburg, Russia, May 25-29, 2015
2015
37.
A. Arnold; M. Ehrhardt
A Transparent Boundary Condition for an Elastic Bottom in Underwater Acoustics
Dimov, I. and Farago, I. and Vulkov, L., Autoren, Finite Difference Methods,Theory and Applications., FDM 2014 Band 9045 aus Lecture Notes in Computer Science , Seite 15--24.
Herausgeber: Springer,
2015
36.
D. Heubes; A. Bartel; M. Ehrhardt
Discrete Artificial Boundary Conditions for the Lattice Boltzmann Method in 2D
ESAIM proceedings of DSFD-2014 Band 52 , Seite 47--65.
2015
35.
D. Heubes; A. Bartel; M. Ehrhardt
Exact Artificial Boundary Conditions for a Lattice Boltzmann Method
Comput. Math. Appl., 67(11):2041--2054
2014
34.
I. Famelis; G. Galanis; M. Ehrhardt; D. Triantafyllou
Classical and Quasi-Newton methods for a Meteorological Parameters Prediction Boundary Value Problem
Appl. Math. Inf. Sci., 8(6):1--11
2014
33.
D. Heubes; A. Bartel; M. Ehrhardt
Characteristic Boundary Conditions in the Lattice Boltzmann Method for Fluid and Gas dynamics
J. Comput. Appl. Math., 262:51--61
2014
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