Finite difference approximations of first derivatives for two-dimensional grid singularities

Explicit finite difference approximations of first derivatives are developed for two-dimensional three-way and five-way grid singularities. The schemes for three-way singularities also apply for six-way singularities. The development objectives are such that the finite difference schemes are non-dissipative and compatible with regular difference approximations for convection dominated phenomena. Special emphasis is given to the directional dependency of numerical phase speeds and stability limits for explicit time integration. For that purpose, free coefficients are used to minimize the isotropy error of the numerical phase velocity. It is shown that numerical phase speeds and stability limits are almost isotropic. Hence, employment of the proposed schemes is nearly independent of the local orientation of the differencing stencil. The time step limits are greater than those of regular finite differences, indicating that the overall time step of a computation is not reduced by employing the proposed schemes. The developed schemes for three-way singularities are of second, fourth and sixth order of accuracy, whereas the five-way difference approximations are second and fourth order accurate, respectively. ectral characteristics, the formal order of accuracy and the stability limits of the proposed schemes imply that they can favorably be used in combination with regular difference approximations of the same order of accuracy on block-structured grids.