Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A scalar hyperbolic equation with GR-type non-linearity   We study a scalar hyperbolic partial differential equation with non-linear
terms similar to those of the equations of general relativity. The equation has
a number of non-trivial analytical solutions whose existence rely on a delicate
balance between linear and non-linear terms. We formulate two classes of
second-order accurate central-difference schemes, CFLN and MOL, for numerical
integration of this equation. Solutions produced by the schemes converge to
exact solutions at any fixed time $t$ when numerical resolution is increased.
However, in certain cases integration becomes asymptotically unstable when $t$
is increased and resolution is kept fixed. This behavior is caused by subtle
changes in the balance between linear and non-linear terms when the equation is
discretized. Changes in the balance occur without violating second-order
accuracy of discretization. We thus demonstrate that a second-order accuracy
and convergence at finite $t$ do not guarantee a correct asymptotic behavior
and long-term numerical stability.
  Accuracy and stability of integration are greatly improved by an exponential
transformation of the unknown variable.

36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
2
The categories of this paper are: math