Analyses of the Dispersion Overshoot and Inverse Dissipation of the High-Order Finite Difference Scheme (Advances in Applied Mathematics and Mechanics, 2021)
Analyses were performed on the dispersion overshoot and inverse dissipationof the high-order finite difference scheme using Fourier and precision analysis.Schemes under discussion included the pointwise- and staggered-grid type, and werepresented in weighted form using candidate schemes with third-order accuracy andthree-point stencil. All of these were commonly used in the construction ofdifference schemes. Criteria for the dispersion overshoot were presented andtheir critical states were discussed. Two kinds of instabilities were studieddue to inverse dissipation, especially those that occur at lower wave numbers.Criteria for the occurrence were presented and the relationship of the twoinstabilities was discussed. Comparisons were made between the analyticalresults and the dispersion/dissipation relations by Fourier transformationof typical schemes. As an example, an application of the criteria was givenfor the remedy of inverse dissipation in Weirs and Martin's third-order scheme.