Finite difference approximations of first derivatives for three-dimensional grid singularities
Explicit finite difference approximations of first derivatives are developed for three-dimensional four-block and six-block grid singularities. The schemes for the four-block singularity are of second, fourth and sixth order of accuracy, whereas the six-block schemes are of second and fourth order. The work extends a recently reported idea of developing schemes for two-dimensional grid singularities to three dimensions. Term matching of Taylor-series expansions of the dependent variable at neighboring points and the requirement for non-dissipative schemes do not provide enough equations to determine all coefficients; the open coefficients are determined through an optimization process to minimize the isotropy error of the numerical phase velocity. Owing to that optimization, the phase velocities and time-step limits for explicit time integration are very isotropic. The spectral characteristics, as dispersion and dissipation property and the time-step limit indicate that the proposed schemes for grid singularities can be well combined with Cartesian schemes of the same order of accuracy on the regular part of a computational mesh. The formal orders of accuracy of the proposed schemes are verified using the three-dimensional linear convection equation. Stability limits for explicit time integration are given.
finite difference grid singularity isotropy error linear wave propagation numerical stability