Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis

People Research Publications Teaching

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Summary

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 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. Our strategies have already been adapted by a Princeton workgroup to systems of wave equations for materials with cracks was also applied to the Numerov scheme for the one-dimensional Schrödinger equation and to (underwater) acoustics.

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. 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.

As well the approach for deriving discrete TBC was extended by Zisowsky to parabolic systems and systems of Schrödinger equations. 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. 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.

Scientific Objectives

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 and Antoine and Besse . 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 (Kohn-Luttinger model of the valence band) and use the discrete TBC for systems of Schrödinger equations from Zisowsky. ( 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. Let us remark that we already analyzed the coupling to different `parabolic' equations in the sea bottom. 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 existing approach from the literature to the two-dimensional wave equation and analyze the resulting IBVP in regard to its well-posedness and stability. Afterwards, in a second step, we will apply the ansatz of Zisowsky 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. 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.

Software

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

References

1. 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)
   
   • Matlab GUI written for numerical experiments can be found here
2. X. Antoine, C. Besse, M. Ehrhardt and P. Klein, Modeling boundary conditions for solving stationary Schrödinger equations, Preprint 10/04, February 2010.
3. 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.
4. A. Arnold, M. Ehrhardt and I. Sofronov, Approximation and fast calculation of non-local boundary conditions for the time-dependent Schrödinger equation, Proceedings of the 15th International Conference on Domain Decomposition Methods, July 21-25, 2003, Berlin, Germany, pp. 141-148.
5. A. Arnold, M. Ehrhardt, M. Schulte and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation on circular domains, Commun. Math. Sci. Vol. 10, Issue 3 (2012), 889-916.
6. Dang Quang A and M. Ehrhardt, Adequate Numerical Solution of Air Pollution problems by Positive Difference schemes on unbounded domains, Math. Comput. Modelling 44 (2006), 834-856.
7. M. Ehrhardt, Discrete Artificial Boundary Conditions, Ph.D. Thesis, Technische Universität Berlin, 2001.
8. M. Ehrhardt and A. Arnold, Discrete Transparent Boundary Conditions for the Schrödinger Equation, Rivista di Mathematica della Universita di Parma 6 (2001), 57-108.
9. M. Ehrhardt, Finite Difference Schemes on unbounded Domains Chapter 8 in: R.E. Mickens (ed.). Applications of Nonstandard Finite Difference Schemes 2, World Scientific, 2005, 343-384.
10. M. Ehrhardt, Discrete Transparent Boundary Conditions for Schrödinger-type equations for non-compactly supported initial data, Appl. Numer. Math., Vol. 58, Issue 5, (2008), 660-673.
11. M. Ehrhardt and A. Arnold, Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations: Fast Calculation and Approximation, Proceedings of the Seventh European Conference on Underwater Acoustics, July 3-8, 2004, Delft, The Netherlands, pp. 9-14.
12. M. Ehrhardt and R.E. Mickens, Solutions to the Discrete Airy Equation: Application to Parabolic Equation Calculations, J. Comput. Appl. Math. 172 (2004), 183-206.
13. M. Ehrhardt and R.E. Mickens, A fast, stable and accurate numerical method for the Black-Scholes equation of American options, Int. J. Theoret. Appl. Finance 11 (2008), 471-501.
14. M. Ehrhardt and C. Zheng, Exact artificial boundary conditions for problems with periodic structures, J. Comput. Phys. 227 (2008), 6877-6894.
15. M. Ehrhardt, H. Han and C. Zheng, Numerical simulation of waves in periodic structures, Commun. Comput. Phys. Vol. 5, Number 5, (2009), 849-872.
16. M. Ehrhardt and A. Zisowsky, Fast Calculation of Energy and Mass preserving solutions of Schrödinger-Poisson systems on unbounded domains, J. Comput. Appl. Math. 187 (2006), 1-28.
17. M. Ehrhardt and A. Zisowsky, Discrete non-local boundary conditions for Split-Step Padé Approximations of the One-Way Helmholtz Equation, J. Comput. Appl. Math. 200 (2007), 471-490.
18. M. Ehrhardt and A. Zisowsky, Nonlocal Boundary Conditions for Higher-Order Parabolic Equations, Proceedings in Applied Mathematics and Mechanics (PAMM) 6 (2006), 733-734.
19. K. Hoke, Diskrete transparente Randbedingungen für hyperbolische Systeme (Discrete transparent Boundary Conditions for hyperbolic Systems, Diploma thesis (in german), September 2006.
20. 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.
21. 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, Journal of Applied Mathematics and Mechanics (ZAMM) 85 (2005), 793-805.
22. A. Zisowsky and M. Ehrhardt, Discrete Transparent Boundary Conditions for Parabolic Systems, Math. Comput. Modelling 43 (2006), 294-309.