How do you do stable simulations at a coordinate singularity?

Spherical coordinates blow up at r=0, which wreaks havoc on numerical simulations of waves and gravitational fields. These new discrete operators are built to satisfy an energy-conservation identity right through that singularity, reaching sixth-order accuracy while placing a grid point exactly at the origin. Demonstrated on the scalar wave equation, the approach offers a cleaner path to stable, high-accuracy simulations of anything from gravitational collapse to stellar oscillations.