Journal of Integral Equations and Applications

Runge-Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equation

Abstract

We propose a numerical scheme to solve the time-dependent linear Schr\"odinger equation. The discretization is carried out by combining a Runge-Kutta time stepping scheme with a finite element discretization in space. Since the Schr\"odinger equation is posed on the whole space $\mathbb{R}^d$, we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper, we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.

Article information

Source
J. Integral Equations Applications, Volume 29, Number 1 (2017), 189-250.

Dates
First available in Project Euclid: 27 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1490583475

Digital Object Identifier
doi:10.1216/JIE-2017-29-1-189

Mathematical Reviews number (MathSciNet)
MR3628111

Zentralblatt MATH identifier
1361.65076

Citation

Melenk, Jens Markus; Rieder, Alexander. Runge-Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equation. J. Integral Equations Applications 29 (2017), no. 1, 189--250. doi:10.1216/JIE-2017-29-1-189. https://projecteuclid.org/euclid.jiea/1490583475

References

• Xavier Antoine, Anton Arnold, Christophe Besse, Matthias Ehrhardt and Achim Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Comm. Comp. Phys. 4 (2008), 729–796.
• A. Bamberger and T. Ha Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d'une onde acoustique, I, Math. Meth. Appl. Sci. 8 (1986), 405–435.
• ––––, Formulation variationnelle pour le calcul de la diffraction d'une onde acoustique par une surface rigide, Math. Meth. Appl. Sci. 8 (1986), 598–608.
• Lehel Banjai, Multistep and multistage convolution quadrature for the wave equation: Algorithms and experiments, SIAM J. Sci. Comp. 32 (2010), 2964–2994.
• Lehel Banjai, Antonio R. Laliena and Francisco-Javier Sayas, Fully discrete Kirchhoff formulas with CQ-BEM, IMA J. Numer. Anal. 35 (2015), 859–884.
• Lehel Banjai and Christian Lubich, An error analysis of Runge-Kutta convolution quadrature, BIT 51 (2011), 483–496.
• Lehel Banjai, Christian Lubich and Jens Markus Melenk, Runge-Kutta convolution quadrature for operators arising in wave propagation, Numer. Math. 119 (2011), 1–20.
• Lehel Banjai, Christian Lubich and Francisco-Javier Sayas. Stable numerical coupling of exterior and interior problems for the wave equation, Numer. Math. 129 (2015), 611–646.
• Lehel Banjai and Stefan Sauter, Rapid solution of the wave equation in unbounded domains, SIAM J. Numer. Anal. 47 (2008/09), 227–249.
• Timo Betcke, Simon Arridge, Joel Phillips, Wojciech Smigaj and Martin Schweiger, Solving boundary integral problems with BEM++, ACM Trans. Math. Software, 2013.
• Philip Brenner and Vidar Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal. 16 (1979), 683–694.
• Martin Costabel, A symmetric method for the coupling of finite elements and boundary elements, in The mathematics of finite elements and applications, VI, Academic Press, London, 1988.
• Michel Crouzeix, Sur les méthodes de Runge Kutta pour l'approximation des problèmes d'évolution, in Computing methods in applied sciences and engineering, Part 1, Lect. Notes Econ. Math. Syst. 134, Springer, Berlin, 1976.
• E. Hairer and G. Wanner, Solving ordinary differential equations, II, Springer Ser. Comp. Math. 14, Springer-Verlag, Berlin, 2010.
• Hou De Han, A new class of variational formulations for the coupling of finite and boundary element methods, J. Comp. Math. 8 (1990), 223–232.
• George C. Hsiao and Wolfgang L. Wendland, Boundary integral equations, Appl. Math. Sci. 164, Springer-Verlag, Berlin, 2008.
• Claes Johnson and J.-Claude Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp. 35 (1980), 1063–1079.
• Antonio R. Laliena and Francisco-Javier Sayas, Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves, Numer. Math. 112 (2009), 637–678.
• Ch. Lubich and A. Ostermann, Runge-Kutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993), 105–131.
• Christian Lubich, Convolution quadrature and discretized operational calculus, I, Numer. Math. 52 (1988), 129–145.
• ––––, Convolution quadrature and discretized operational calculus, II, Numer. Math. 52 (1988), 413–425.
• William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
• Michael Reed and Barry Simon, Methods of modern mathematical physics, I, second edition, Academic Press, Inc., New York, 1980.
• Stefan A. Sauter and Christoph Schwab, Boundary element methods, Springer Ser. Comp. Math. 39, Springer-Verlag, Berlin, 2011.
• Achim Schädle, Non-reflecting boundary conditions for the two-dimensional Schrödinger equation, Wave Motion 35 (2002), 181–188.
• Joachim Schöberl, Ngsolve finite element library, sourceforge.net/projects/ngsolve, 2015.
• E.M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.
• Olaf Steinbach, Numerical approximation methods for elliptic boundary value problems, Springer, New York, 2008.
• Dirk Werner, Funktionalanalysis, Springer-Verlag, Berlin, 2000.