Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Error estimates of local energy regularization for the logarithmic
  Schrodinger equation   The logarithmic nonlinearity has been used in many partial differential
equations (PDEs) for modeling problems in various applications.Due to the
singularity of the logarithmic function, it introducestremendous difficulties
in establishing mathematical theories, as well asin designing and analyzing
numerical methods for PDEs with such nonlinearity. Here we take the logarithmic
Schr\"odinger equation (LogSE)as a prototype model. Instead of regularizing
$f(\rho)=\ln \rho$ in theLogSE directly and globally as being done in the
literature, we propose a local energy regularization (LER) for the LogSE
byfirst regularizing $F(\rho)=\rho\ln \rho -\rho$ locally near $\rho=0^+$ with
a polynomial approximation in the energy functional of the LogSE and then
obtaining an energy regularized logarithmic Schr\"odinger equation (ERLogSE)
via energy variation. Linear convergence is established between the solutions
of ERLogSE and LogSE in terms of a small regularization parameter $0<\ep\ll1$.
Moreover, the conserved energy of the ERLogSE converges to that of LogSE
quadratically, which significantly improvesthe linear convergence rate of the
regularization method in the literature. Error estimates are alsopresented for
solving the ERLogSE by using Lie-Trotter splittingintegrators. Numerical
results are reported to confirm our errorestimates of the LER and of the
time-splitting integrators for theERLogSE. Finally our results suggest that the
LER performs better than regularizing the logarithmic nonlinearity in the LogSE
directly.

22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph