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
- 1991
368.
Fink, Ewald H.; Setzer, Klaus-Dieter; Ramsay, D. A.; Vervloet, M.
A new band spectrum of BiO in the near-infrared region
Chemical Physics Letters, 179 (1-2) :103-108
1991367.
Becker, Eilhard; Benter, Thorsten; Kampf, R.; Schindler, Ralph N.; Wille, Uta
A Redetermination of the Rate Constant of the Reaction F + HNO\(_{3}\) → HF + NO\(_{3}\)
Berichte der Bunsengesellschaft für physikalische Chemie, 95 (10) :1168-1173
1991366.
Becker, Eilhard; Benter, Thorsten; Kampf, R.; Schindler, Ralph N.; Wille, Uta
A Redetermination of the Rate Constant of the Reaction F + HNO\(_{3}\) → HF + NO\(_{3}\)
Berichte der Bunsengesellschaft für physikalische Chemie, 95 (10) :1168-1173
1991365.
Becker, Eilhard; Benter, Thorsten; Kampf, R.; Schindler, Ralph N.; Wille, Uta
A Redetermination of the Rate Constant of the Reaction F + HNO3 → HF + NO3
Berichte der Bunsengesellschaft für physikalische Chemie, 95 (10) :1168-1173
1991364.
Jensen, Per; Bunker, Philip R.; Karpfen, Alfred
An ab initio calculation of the nonadiabatic effect on the tunneling splitting in vibrationally excited (HF)\(_{2}\)
Journal of Molecular Spectroscopy, 148 (2) :385-390
1991363.
Jensen, Per; Bunker, Philip R.; Karpfen, Alfred
An ab initio calculation of the nonadiabatic effect on the tunneling splitting in vibrationally excited (HF)\(_{2}\)
Journal of Molecular Spectroscopy, 148 (2) :385-390
1991362.
Jensen, Per; Bunker, Philip R.; Karpfen, Alfred
An ab initio calculation of the nonadiabatic effect on the tunneling splitting in vibrationally excited (HF)2
Journal of Molecular Spectroscopy, 148 (2) :385-390
1991361.
Jensen, Per; Marshall, Mark D.; Bunker, Philip R.; Karpfen, Alfred
An ab initio close-coupling calculation of the lower vibrational energies of the HCl dimer
Chemical Physics Letters, 180 (6) :594-600
1991360.
Jensen, Per; Marshall, Mark D.; Bunker, Philip R.; Karpfen, Alfred
An ab initio close-coupling calculation of the lower vibrational energies of the HCl dimer
Chemical Physics Letters, 180 (6) :594-600
1991359.
Jensen, Per; Marshall, Mark D.; Bunker, Philip R.; Karpfen, Alfred
An ab initio close-coupling calculation of the lower vibrational energies of the HCl dimer
Chemical Physics Letters, 180 (6) :594-600
1991358.
Marshall, Mark D.; Jensen, Per; Bunker, Philip R.
An ab initio close-coupling calculation of the lower vibrational energies of the HF dimer
Chemical Physics Letters, 176 (3-4) :255-260
1991357.
Marshall, Mark D.; Jensen, Per; Bunker, Philip R.
An ab initio close-coupling calculation of the lower vibrational energies of the HF dimer
Chemical Physics Letters, 176 (3-4) :255-260
1991356.
Marshall, Mark D.; Jensen, Per; Bunker, Philip R.
An ab initio close-coupling calculation of the lower vibrational energies of the HF dimer
Chemical Physics Letters, 176 (3-4) :255-260
1991355.
Karpfen, Alfred; Bunker, Philip R.; Jensen, Per
An ab initio study of the hydrogen chloride dimer: the potential energy surface and the characterization of the stationary points
Chemical Physics, 149 (3) :299-309
1991354.
Karpfen, Alfred; Bunker, Philip R.; Jensen, Per
An ab initio study of the hydrogen chloride dimer: the potential energy surface and the characterization of the stationary points
Chemical Physics, 149 (3) :299-309
1991353.
Karpfen, Alfred; Bunker, Philip R.; Jensen, Per
An ab initio study of the hydrogen chloride dimer: the potential energy surface and the characterization of the stationary points
Chemical Physics, 149 (3) :299-309
1991352.
Bunker, Philip R.; Epa, V. C.; Jensen, Per; Karpfen, Alfred
An analytical ab initio potential surface and the calculated tunneling energies for the HCl dimer
Journal of Molecular Spectroscopy, 146 (1) :200-219
1991351.
Bunker, Philip R.; Epa, V. C.; Jensen, Per; Karpfen, Alfred
An analytical ab initio potential surface and the calculated tunneling energies for the HCl dimer
Journal of Molecular Spectroscopy, 146 (1) :200-219
1991350.
Bunker, Philip R.; Epa, V. C.; Jensen, Per; Karpfen, Alfred
An analytical ab initio potential surface and the calculated tunneling energies for the HCl dimer
Journal of Molecular Spectroscopy, 146 (1) :200-219
1991349.
Fink, Ewald H.; Setzer, Klaus-Dieter; Wildt, J{ü}rgen; Ramsay, D. A.; Vervloet, M.
Collision-induced emission of O\(_{2}\)(b\(^{1}\)\(\Sigma\)\(_{g}\)\(^{+}\) → a\(^{1}\)\(\Delta\)\(_{g}\)) in the gas phase
International Journal of Quantum Chemistry, 39 (3) :287-298
1991348.
Fink, Ewald H.; Setzer, Klaus-Dieter; Wildt, J{ü}rgen; Ramsay, D. A.; Vervloet, M.
Collision-induced emission of O\(_{2}\)(b\(^{1}\)\(\Sigma\)\(_{g}\)\(^{+}\) → a\(^{1}\)\(\Delta\)\(_{g}\)) in the gas phase
International Journal of Quantum Chemistry, 39 (3) :287-298
1991347.
Fink, Ewald H.; Setzer, Klaus-Dieter; Wildt, Jürgen; Ramsay, D. A.; Vervloet, M.
Collision-induced emission of O2(b1Σg+ → a1Δg) in the gas phase
International Journal of Quantum Chemistry, 39 (3) :287-298
1991346.
Vilesov, A. F.; Wildt, J{ü}rgen; Fink, Ewald H.
Emission of Xe: N(\(^{2}\)P) collision complexes near the N(\(^{2}\)P→\(^{2}\)D) lines
Chemical Physics, 153 (3) :531-537
1991345.
Vilesov, A. F.; Wildt, J{ü}rgen; Fink, Ewald H.
Emission of Xe: N(\(^{2}\)P) collision complexes near the N(\(^{2}\)P→\(^{2}\)D) lines
Chemical Physics, 153 (3) :531-537
1991344.
Vilesov, A. F.; Wildt, Jürgen; Fink, Ewald H.
Emission of Xe: N(2P) collision complexes near the N(2P→2D) lines
Chemical Physics, 153 (3) :531-537
1991
