Illustration of the computation at singularity points. (a) Grid lines near a positive singularity. (b) The difference stencil in the computational domain, with a stencil length of 7. (c) Grid lines near a negative singularity. In (a) and (c), the blue grid lines indicate that the difference stencil requires the use of the singularity, while the red grid lines represent the directions where the stencil is applied normally. (d) and (e) show the specific points needed to construct the difference stencils at positive and negative singularities, respectively. The tables also illustrate part of the difference stencils that require the use of the singularity grid points, where ‘r’ represents completing the stencil along the red grid lines, and ‘b’ represents completing the stencil along the blue grid lines.