Transparent boundary conditons for time-dependent problems

Please always quote using this URN: urn:nbn:de:0297-zib-9558

A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions. Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients.

Metadaten
Author:	Daniel Ruprecht, Achim Schädle, Frank Schmidt, Lin Zschiedrich
Document Type:	ZIB-Report
Tag:	Schrödinger equation; drift diffusion equation; non-reflecting boundary condition; pole condition; transparent boundary condition; wave equation
MSC-Classification:	30-XX FUNCTIONS OF A COMPLEX VARIABLE (For analysis on manifolds, see 58-XX) / 30Exx Miscellaneous topics of analysis in the complex domain / 30E10 Approximation in the complex domain
	65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M20 Method of lines
	65-XX NUMERICAL ANALYSIS / 65Mxx Partial differential equations, initial value and time-dependent initial- boundary value problems / 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Date of first Publication:	2007/05/22
Series (Serial Number):	ZIB-Report (07-12)
ZIB-Reportnumber:	07-12
Published in:	Appeared in: SIAM J. Sci. Comput., Vol. 30 (5) , 2358-2385 (2008)

Open Access