Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Numerical Stability and Catalan Numbers   To predict allowable time-step size for the fully discretized nonlinear
differential equations, a stability theory is developed using exact
determination of an infinite perturbation series. Mathematical induction is
used to determine the coefficients of the series. It is discovered that the
closed-form equation for the nonlinear shift of generic polynomial
non-linearity can be written as a series expansion where the coefficients are
the Pfaff-Fuss-Catalan numbers in Combinatorics. This reveals criteria which
can be used to analytically determine the allowable time step. It is shown that
stability region decreases when the nonlinearity of the differential equation
increases. Therefore, the maximum allowable time step is severely limited by
the nonlinearity even if an unconditionally stable scheme (in a linear sense)
is used. The theory is applied to general system of time-dependent nonlinear
Partial Differential Equations.

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The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: stat
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga