Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Analysis and development of compact finite difference schemes with
  optimized numerical dispersion relations   Finite difference approximation, in addition to Taylor truncation errors,
introduces numerical dispersion-and-dissipation errors into numerical solutions
of partial differential equations. We analyze a class of finite difference
schemes which are designed to minimize these errors (at the expense of formal
order of accuracy), and we analyze the interplay between the Taylor truncation
errors and the dispersion-and-dissipation errors during mesh refinement. In
particular, we study the numerical dispersion relation of the fully discretized
non-dispersive transport equation in one and two space dimensions. We derive
the numerical phase error and the $L^2$-norm error of the solution in terms of
the dispersion-and-dissipation error. Based on our analysis, we investigate the
error dynamics among various optimized compact schemes and the unoptimized
higher-order generalized Pad\'e compact schemes. The dynamics shed light on the
principles of designing suitable optimized compact schemes for a given problem.
Using these principles as guidelines, we then propose an optimized scheme that
prescribes the numerical dispersion relation before finding the corresponding
discretization. This approach produces smaller numerical
dispersion-and-dissipation errors for linear and nonlinear problems, compared
with the unoptimized higher-order compact schemes and other optimized schemes
developed in the literature. Finally, we discuss the difficulty of developing
an optimized composite boundary scheme for problems with non-trivial boundary
conditions. We propose a composite scheme that introduces a buffer zone to
connect an optimized interior scheme and an unoptimized boundary scheme. Our
numerical experiments show that this strategy produces small $L^2$-norm error
when a wave packet passes through the non-periodic boundary.

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The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
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The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
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The categories of this paper are: funct-an