Analyses were performed on the dispersion overshoot and inverse dissipation
of the high-order finite difference scheme using Fourier and precision analysis.
Schemes under discussion included the pointwise- and staggered-grid type, and were
presented in weighted form using candidate schemes with third-order accuracy and
three-point stencil. All of these were commonly used in the construction of
difference schemes. Criteria for the dispersion overshoot were presented and
their critical states were discussed. Two kinds of instabilities were studied
due to inverse dissipation, especially those that occur at lower wave numbers.
Criteria for the occurrence were presented and the relationship of the two
instabilities was discussed. Comparisons were made between the analytical
results and the dispersion /dissipation relations by Fourier transformation
of typical schemes. As an example, an application of the criteria was given
for the remedy of inverse dissipation in Weirs and Martin's third-order scheme.