Numerical Treatment of PDEs on unbounded Domains

(Project of Junior Research Group on Applied Analysis)

(Deutsche Version)

Description

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). Some Review articles about computation on unbounded domainscan be found here. We remark that TBCs are also applied succesfully in optimal boundary control for solving inverse problems [MeHe].

This project is 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. Our strategies have already been adapted by a Princeton workgroup to systems of wave equations for materials with cracks [WEHu], was also applied to the Numerov scheme in [Moy1], [Moy2] for the one-dimensional Schrödinger equation and to (underwater) acoustics in [Mik]. See also Section 2.3 in [CRS].

Another problem is the high numerical effort. Since the discrete TBC includes a convolution w.r.t. 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. On a continuous level this idea is related to [AGH], [GrKe] and [Sof]. We also prove stability criteria of the resulting initial boundary value problem and present an error analysis of the resulting numerical scheme. Numerical examples show the efficiency of the proposed method on several examples (cf. [AES]).

As well the approach for deriving discrete TBC was extended by Zisowsky to parabolic systems and systems of Schrödinger equations [Zis]. Especially for this enormous costly boundary conditions the above approximation ansatz is very profitable.

Furthermore, the time-dependent Schrödinger-Poisson System in radial coordinates was investigated numerically in collaboration with J. Soler and E. Ruiz Arriola. For the unitary and unconditional stable Crank-Nicolson scheme discrete TBCs were incorporated into an existing Fortran code to improve the long-time simulation of the whole space evolution in the spherically symmetric case. Special attention was paid to the study of the asymptotic behaviour in time for the solutions in the attractive case with positive energy. Several test examples were done to numerically check the validity of a dispersion equation relating density and linear moment dispersion.

Research program

In this project we want to extend the above described approach to Schrödinger-type equations. Thereby we want to derive discrete TBCs for new numerical methods (e.g. a mass and energy conserving predictor-corrector scheme), space-dependent potentials and in the case when the support of the initial data is not completely contained in the computational domain. The results shall be compared with the alternative approaches (e.g. the Pole condition) of F. Schmidt [Sch] and Antoine and Besse [AnBe]. This comparison study is a joint work with the members of project D9. As concrete application in the nanotechnology we collaborate with A. Arnold and Th. Koprucki and compute numerically solutions to the ``double-barrier stepped-quantum-well'' (tunnel structure). Here we consider special systems of Schrödinger equations [BKKR] (Kohn-Luttinger model of the valence band) and use the discrete TBC for systems of Schrödinger equations from [Zis] ( details). As a second application and to demonstrate the effectiveness of the approximative discrete TBC we simulate (with A. Arnold and O. Vanbésien) the transient behaviour of quantum wave guides (``multi-channel quantum waveguides'') in 2D.

Closely connected with the derivation of TBCs for different equation is the analysis of initial boundary value problems (IBVPs) that arise when coupling different models. One can find applications for (discrete) wave equation models e.g. in underwater acoustics. There one wants to couple a fluid model for the water with an elastic model for the sea bottom. In [ArEh] we already analyzed the coupling to different `parabolic' equations in the sea bottom. Jointly with M. Baro and H.-C. Kaiser from project D4 we will analyze the well-posedness and stability (discrete) Initial boundary value problems arising from a Quantum-Drift-Diffusion equation. Furthermore we will investigate the application of TBCs (Coupling to the Schrödinger equation).

As a third focus we want to develop discrete TBCs for hyperbolic systems. First we carry over the approach from [4],[AES] to the two-dimensional wave equation and analyze the resulting IBVP in regard to its well-posedness and stability. The numerical results shall be compared with those of Alpert, Greengard and Hagstrom [AGH]. Afterwards, in a second step, we will apply the ansatz of Zisowsky [Zis] to hyperbolic systems. In this connection we want to study the (discrete) coupling of different hyperbolic systems. The numerical results in the application to the Euler equations of gas dynamics shall be compared to that of Rowley and Colonius [RoCo]. As a concrete application we use the discrete TBCs in the simulation of quantum devices with classical regions that can be described by hydrodynamical models. We will consider the classical Baccarani-Wordeman-Model (as it is used in industrial codes) and the improved hydrodynamic model for semiconductors.

Finally, to illustrate the broad range of applicability of our approach we derive efficient discrete artificial boundary conditions for the Black-Scholes equation of American options [EhMi2]. Here we are in discussion with project E6.

To acquire young scientific researchers we read during the winter term 2003/04 (after the lecture Numerik partieller Differentialgleichungen(Numerics of Partial Differential Equations) in the summer term 03) the lecture Theorie und Numerik hyperbolischer Erhaltungsgleichungen (Theory and Numerics of Hyperbolic Conservation Laws) which will be followed by the seminar Partielle Differentialgleichungen in der Angewandten Mathematik (Partial Differential Equations in Applied Mathematics). In the winter term 2004/05 we offer the lecture and exercise Asymptotische Analysis(Asymptotical Analysis) and afterwards in the summer term 2005 we read the special lecture Modellierung, Analysis und Numerik in der Halbleiter-Technologie (Modelling, Analysis and Numerics in the Semiconductors Technology).

Software

Our approach was implemented by C.A. Moyer in the QMTools software package for quantum mechanical applications (cf. [Moy]).

References

[AGH]	B.K. Alpert, L. Greengard and T. Hagstrom,
	Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal. 37 (2000), 1138-1164.
[AMRe]	I. Alonso-Mallo and N. Reguera,
	Weak ill-posedness of spatial discretizations of absorbing boundary conditions for Schrödinger-type equations, SIAM J. Numer. Anal. 40 (2002), 134-158.
[AnBe]	X. Antoine and C. Besse,
	Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation, J. Comp. Phys. 188 (2003), 157-175.
[ABM]	X. Antoine, C. Besse, and V. Mouysset,
	Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions, Math. Comp. 73 (2004), 1779-1799.
[ApKr]	D. Appelö and G. Kreiss,
	Stabilized local nonreflecting boundary conditions for high order centered schemes for the wave equation, Technical report, NADA, Royal Institute of Technology, Sweden, 2003.
[Arn]	A. Arnold,
	Numerically Absorbing Boundary Conditions for Quantum Evolution Equations, VLSI Design 6 (1998), 313-319.
[ArEh]	A. Arnold and M. Ehrhardt,
	Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics, J. Comp. Phys. 145 (1998), 611-638.
[ArSc07]	A. Arnold and M. Schulte,
	Transparent boundary conditions for quantum-waveguide simulations, to appear in Mathematics and Computers in Simulation (2007), Proceedings of MATHMOD 2006, Vienna, Austria.
[ArSc08]	A. Arnold and M. Schulte,
	Discrete transparent boundary conditions for the Schrödinger equation - a compact higher order scheme, to appear in Kinetic and Related Models 1, 2008.
[BKKR]	U. Bandelow, H.-C. Kaiser, Th. Koprucki and J. Rehberg,
	Spectral properties of k-p Schrödinger operators in one space dimension, Numer. Funct. Anal. Optimization 21 (2000), 379-409.
[CRS]	N. Carjan, M. Rizea and D. Strottman,
	Improved boundary conditions for the decay of low lying metastable proton states in a time-dependent approach, Comput. Phys. Commun. 173 (2005), 41-60.
[DuZl]	B. Ducomet and A. Zlotnik,
	On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part I, Commun. Math. Sci. 4 (2006), 741-766.
[WEHu]	W. E and Z. Huang
	A dynamic atomistic-continuum method for the simulation of crystalline materials, J. Comp. Phys. 182 (2002), 234-261.
[Ehr]	M. Ehrhardt,
	Discrete Artificial Boundary Conditions, Ph.D. Thesis, Technische Universität Berlin, 2001.
[EhAr]	M. Ehrhardt and A. Arnold,
	Discrete Transparent Boundary Conditions for the Schrödinger Equation, Rivista di Mathematica della Universita di Parma, Volume 6, Number 4 (2001), 57-108.
[Gal]	H. Galicher,
	Transparent Boundary Conditions for the one-dimensional Schrödinger Equation with periodic potential at infinity, submitted to: Commun. Math. Sci., 2007.
[GrKe]	M.J. Grote and J.B. Keller,
	Exact nonreflecting boundary conditions for the time dependent wave equation, SIAM J. Appl. Math. 55 (1995), 280-297.
[JiGr]	S. Jiang and L. Greengard,
	Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension, Comp. Math. Appl. 47 (2004), 955-966.
[MeHe]	M. Meyer and J.P. Hermand,
	Optimal nonlocal boundary control of the wide-angle parabolic equation for inversion of a waveguide acoustic field, J. Acous. Soc. Am. 117 (2005), 2937-2948.
[Mik]	D. Mikhin,
	Exact discrete nonlocal boundary conditions for high-order Padé parabolic equations, J. Acous. Soc. Am. 116 (2004), 2864-2875.
[Moy1]	C.A. Moyer,
	Numerov Extension of Transparent Boundary Conditions for the Schrödinger Equation in One Dimension, Amer. J. Phys. 72 (2004), 351-358.
[Moy2]	C.A. Moyer,
	Numerical solution of the stationary state Schrödinger Equation using discrete transparent boundary conditions, Computing in Science & Engineering 8 (2006), 32 - 40.
[RoCo]	C.W. Rowley and T. Colonius,
	Discretely nonreflecting boundary conditions for linear hyperbolic systems, J. Comp. Phys. 145 (2000), 500-538.
[RSSZ]	D. Ruprecht, A. Schädle, F. Schmidt and L. Zschiedrich,
	Transparent boundary conditions for time-dependent problems, ZIB-Report 07-12, ZIB Berlin, May 2007.
[Sch]	F. Schmidt,
	A New Approach to Coupled Interior-Exterior Helmholtz-Type Problems: Theory and Algorithms, Habilitation, Freie Universität Berlin, 2002.
[Sch04]	M. Schulte,
	Transparente Randbedingungen für die Schrödinger-Gleichung, Diploma Thesis, Universität Münster, March 2004.
[Sch07]	M. Schulte,
	Numerical Solution of the Schrödinger Equation on unbounded Domains, Dissertation, Universität Münster, 2007.
[Sof]	I. Sofronov,
	Conditions for Complete Transparency on the Sphere for the Three-Dimensional Wave Equation Fast calculation, approximation, and stability, Russian Acad. Sci. Dokl. Math. 46 (1993), 397-401.

Publications of the Project

[DTBC]	X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle,
	A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations, Commun. Comput. Phys. 4 (2008), 729-796. (open-access article) There is a Matlab GUI written for numerical experiments.
[AES]	A. Arnold, M. Ehrhardt and I. Sofronov,
	Discrete Transparent Boundary Conditions for the Schrödinger Equation: Fast calculation, approximation, and stability, Comm. Math. Sci. 1 (2003), 501-556.
[AES:P]	A. Arnold, M. Ehrhardt and I. Sofronov,
	Approximation and fast calculation of non-local boundary conditions for the time-dependent Schrödinger equation, ( PDF-Format), Proceedings of the 15th International Conference on Domain Decomposition Methods, July 21-25, 2003, Berlin, Germany, pp. 141-148.
[AESS]	A. Arnold, M. Ehrhardt, M. Schulte and I. Sofronov,
	Discrete Transparent Boundary Conditions for the two-dimensional Schrödinger equation, ( PDF-Format), Matheon-Preprint No. 332, May 2006.
[DaEh]	Dang Quang A and M. Ehrhardt,
	Adequate Numerical Solution of Air Pollution problems by Positive Difference schemes on unbounded domains, ( PDF-Format), Math. Comput. Modelling 44 (2006), 834-856.
[Ehr1]	M. Ehrhardt,
	Finite Difference Schemes on unbounded Domains, ( PDF-Format), Chapter 8 in: R.E. Mickens (ed.), Applications of Nonstandard Finite Difference Schemes, volume 2, World Scientific, 2005, pp. 343-384.
[Ehr2]	M. Ehrhardt,
	Discrete Transparent Boundary Conditions for Schrödinger-type equations for non-compactly supported initial data, ( PDF-Format), Appl. Numer. Math., Vol. 58, Issue 5, (2008), 660-673.
[EhAr:P]	M. Ehrhardt and A. Arnold,
	Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations: Fast Calculation and Approximation, ( PDF-Format), Proceedings of the Seventh European Conference on Underwater Acoustics, July 3-8, 2004, Delft, The Netherlands, pp. 9-14.
[EhMi1]	M. Ehrhardt and R.E. Mickens,
	Solutions to the Discrete Airy Equation: Application to Parabolic Equation Calculations , ( PDF-Format), J. Comput. Appl. Math. 172 (2004), 183-206.
[EhMi2]	M. Ehrhardt and R.E. Mickens,
	A fast, stable and accurate numerical method for the Black-Scholes equation of American options, ( PDF-Format), Int. J. Theoret. Appl. Finance 11 (2008), 471-501.
[EhZh]	M. Ehrhardt and C. Zheng,
	Exact artificial boundary conditions for problems with periodic structures, J. Comput. Phys. Vol. 227, Issue 14, (2008), 6877-6894.
[EHZ]	M. Ehrhardt, H. Han and C. Zheng,
	Numerical simulation of waves in periodic structures, Commun. Comput. Phys. Vol. 5, Number 5, (2009), 849-872.
[EhZi1]	M. Ehrhardt and A. Zisowsky,
	Fast Calculation of Energy and Mass preserving solutions of Schrödinger-Poisson systems on unbounded domains, ( PDF-Format), J. Comput. Appl. Math. 187 (2006), 1-28.
[EhZi2]	M. Ehrhardt and A. Zisowsky,
	Discrete non-local boundary conditions for Split-Step Padé Approximations of the One-Way Helmholtz Equation, ( PDF-Format), J. Comput. Appl. Math. 200 (2007), 471-490.
[EhZi:P]	M. Ehrhardt and A. Zisowsky,
	Nonlocal Boundary Conditions for Higher-Order Parabolic Equations, ( PDF-Format), Proceedings in Applied Mathematics and Mechanics (PAMM), Vol. 6, Issue 1 (2006), 733--734 (Special Issue: GAMM Annual Meeting 2006).
[Hok06]	K. Hoke,
	Diskrete transparente Randbedingungen für hyperbolische Systeme (Discrete transparent Boundary Conditions for hyperbolic Systems, Diploma thesis (in german), September 2006 work presented at the student conference Dies Mathematicus, TU Berlin, October 27, 2006.
[SuEh07]	L. Šumichrast and M. Ehrhardt,
	On the three formulations of transparent boundaries for the beam propagation method, Proceedings of AMTEE 2007 - Advanced Methods of the Theory of Electrical Engineering (Applied to Power Systems), Pilsen, Czech Republic, September, 10-12, 2007.
[Zis]	A. Zisowsky,
	Discrete Transparent Boundary Conditions for Systems of Evolution Equations (pdf), (abstract), Ph.D. Thesis, Technische Universität Berlin, July 2003.
[ZAEK1]	A. Zisowsky, A. Arnold, M. Ehrhardt and Th. Koprucki,
	Discrete Transparent Boundary Conditions for Time-Dependent Systems of Schrödinger Equations, ( PDF-Format), Matheon-Preprint No. 104, March 2004.
[ZAEK2]	A. Zisowsky, A. Arnold, M. Ehrhardt and Th. Koprucki,
	Discrete Transparent Boundary Conditions for Time-Dependent Systems of Schrödinger Equations with Application to Quantum-Heterostructures, ( PDF-Format), Journal of Applied Mathematics and Mechanics (ZAMM),Vol. 85, No. 11, (2005), 793-805.
[ZiEh]	A. Zisowsky and M. Ehrhardt,
	Discrete Transparent Boundary Conditions for Parabolic Systems, ( PDF-Format), Math. Comput. Modelling 43 (2006), 294-309.

Guests

July 13-14, 2003	A. Arnold, Institute for Analysis and Scientific Computing Technische Universität Wien, Austria.
May 9-31, 2004	I. Sofronov, Keldysh Institute of Applied Mathematics, Moscow, Russia
May 9-11, 2004	A. Arnold, Institute for Analysis and Scientific Computing Technische Universität Wien, Austria.
January 26- February 7, 2005	Eli Turkel, Department of Applied Mathematics at School of Mathematical Sciences, Tel Aviv University, Israel.
June 22- July 14, 2005	Dang Quang A, Department of Mathematical Methods for Information Technology Institute of Information Technology, Hanoi, Vietnam.
September 24 - October 9, 2005	I. Sofronov, Keldysh Institute of Applied Mathematics, Moscow, Russia
February 12 - March 26, 2006	M. Schulte, Universität Münster, Germany.
March 23 - 31, 2006	I. Sofronov, Keldysh Institute of Applied Mathematics, Moscow, Russia

Links

	Tom Hagstrom's Radiation Boundary Condition Page
	Numerical Methods for Wave Propagation on Unbounded Domains, Mini-Symposia at 1996 SIAM Annual Meeting Kansas City, Missouri July 22 -- 26, 1996
	Domain Decomposition Methods for Wave Propagation in Unbounded Media, Minisymposium at the 15th International Conference on Domain Decomposition Methods, July 21 - 25, FU Berlin, Germany.
	6th International Congress on Industrial and Applied Mathematics, Zürich, Switzerland, July 16-20, 2007. Minisymposium Artificial Boundary Conditions for linear and nonlinear Schrödinger equations

ehrhardt@math.tu-berlin.de