TY - JOUR
T1 - Finite difference solutions of the nonlinear Schrödinger equation and their conservation of physical quantities
AU - Heitzinger, Clemens
AU - Ringhofer, Christian
AU - Selberherr, Siegfried
PY - 2007
Y1 - 2007
N2 - The solutions of the nonlinear Schrödinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest being the local norm square conservation law. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we first give the conservation laws that can be calculated by means of Lie theory and then critically compare the quality of different finite difference methods that have been proposed in geometric integration with respect to conservation laws. We find that finite difference schemes derived by writing the Schrödinger dinger equation as an (artificial) Hamiltonian system do not necessarily conserve important physical quantities better than other methods.
AB - The solutions of the nonlinear Schrödinger equation are of great importance for ab initio calculations. It can be shown that such solutions conserve a countable number of quantities, the simplest being the local norm square conservation law. Numerical solutions of high quality, especially for long time intervals, must necessarily obey these conservation laws. In this work we first give the conservation laws that can be calculated by means of Lie theory and then critically compare the quality of different finite difference methods that have been proposed in geometric integration with respect to conservation laws. We find that finite difference schemes derived by writing the Schrödinger dinger equation as an (artificial) Hamiltonian system do not necessarily conserve important physical quantities better than other methods.
KW - Conservation law
KW - Finite difference scheme
KW - Nonlinear Schrödinger equation
KW - Variational symmetry
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U2 - 10.4310/CMS.2007.v5.n4.a2
DO - 10.4310/CMS.2007.v5.n4.a2
M3 - Article
AN - SCOPUS:39049130341
SN - 1539-6746
VL - 5
SP - 779
EP - 788
JO - Communications in Mathematical Sciences
JF - Communications in Mathematical Sciences
IS - 4
ER -