Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics

The proposed scheme, which is a conservative form of the interpolated differential operator scheme (IDO-CF), can provide high accurate solutions for both compressible and incompressible fluid equations. Spatial discretizations with fourth-order accuracy are derived from interpolation functions locally constructed by both cell-integrated values and point values. These values are coupled and time-integrated by solving fluid equations in the flux forms for the cell-integrated values and in the derivative forms for the point values. The IDO-CF scheme exactly conserves mass, momentum, and energy, retaining the high resolution more than the non-conservative form of the IDO scheme. A direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods. Benchmark tests of Riemann problems and lid-driven cavity flows show that the IDO-CF scheme is immensely promising in compressible and incompressible fluid dynamics studies.